Practice sheet

Fresh questions from the whole chapter, to do on paper.

New questions Include answers

To save a PDF, choose “Save as PDF” in the print dialog.

Evaluate $\log_{ 4 } 16$ .
Solve $6^x = 21$ . Write the exact answer using a logarithm.
Match each logarithm statement with the same statement written as a power.
1. $\log_{ 8 } y = 2$
2. $\log_{ 8 } y = 3$
3. $\log_{ 8 } y = -1$
4. $\log_{ 8 } y = \tfrac{1}{2}$
1. $y = 8^{2}$
2. $y = 8^{3}$
3. $y = \dfrac{1}{ 8 }$
4. $y = \sqrt{ 8 }$
Match each logarithm statement with the same statement written as a power.
1. $\log_{ 3 } y = 2$
2. $\log_{ 3 } y = 3$
3. $\log_{ 3 } y = -1$
4. $\log_{ 3 } y = \tfrac{1}{2}$
1. $y = 3^{2}$
2. $y = 3^{3}$
3. $y = \dfrac{1}{ 3 }$
4. $y = \sqrt{ 3 }$
Match each logarithm statement with the same statement written as a power.
1. $\log_{ 4 } y = 2$
2. $\log_{ 4 } y = 3$
3. $\log_{ 4 } y = -1$
4. $\log_{ 4 } y = \tfrac{1}{2}$
1. $y = 4^{2}$
2. $y = 4^{3}$
3. $y = \dfrac{1}{ 4 }$
4. $y = \sqrt{ 4 }$
Match each logarithm statement with the same statement written as a power.
1. $\log_{ 6 } y = 2$
2. $\log_{ 6 } y = 3$
3. $\log_{ 6 } y = -1$
4. $\log_{ 6 } y = \tfrac{1}{2}$
1. $y = 6^{2}$
2. $y = 6^{3}$
3. $y = \dfrac{1}{ 6 }$
4. $y = \sqrt{ 6 }$
Match each logarithm statement with the same statement written as a power.
1. $\log_{ 2 } y = 2$
2. $\log_{ 2 } y = 3$
3. $\log_{ 2 } y = -1$
4. $\log_{ 2 } y = \tfrac{1}{2}$
1. $y = 2^{2}$
2. $y = 2^{3}$
3. $y = \dfrac{1}{ 2 }$
4. $y = \sqrt{ 2 }$
Match each logarithm statement with the same statement written as a power.
1. $\log_{ 9 } y = 2$
2. $\log_{ 9 } y = 3$
3. $\log_{ 9 } y = -1$
4. $\log_{ 9 } y = \tfrac{1}{2}$
1. $y = 9^{2}$
2. $y = 9^{3}$
3. $y = \dfrac{1}{ 9 }$
4. $y = \sqrt{ 9 }$
Match each logarithm statement with the same statement written as a power.
1. $\log_{ 5 } y = 2$
2. $\log_{ 5 } y = 3$
3. $\log_{ 5 } y = -1$
4. $\log_{ 5 } y = \tfrac{1}{2}$
1. $y = 5^{2}$
2. $y = 5^{3}$
3. $y = \dfrac{1}{ 5 }$
4. $y = \sqrt{ 5 }$
Match each logarithm statement with the same statement written as a power.
1. $\log_{ 7 } y = 2$
2. $\log_{ 7 } y = 3$
3. $\log_{ 7 } y = -1$
4. $\log_{ 7 } y = \tfrac{1}{2}$
1. $y = 7^{2}$
2. $y = 7^{3}$
3. $y = \dfrac{1}{ 7 }$
4. $y = \sqrt{ 7 }$
The graph of $y = 3\cdot 4^x + 7$ has a horizontal asymptote $y = c$ . What is $c$ ?
The graph of $y = \log_{ 4 }(x - 6) + 1$ has a vertical asymptote $x = c$ . What is $c$ ?
Draw the graph of $y = 1 \cdot 2^x + 3$ : place two points and the asymptote.
A sample of 64 mg of a radioactive substance has a half-life of 2 years. How many mg are left after 6 years?
A dose of 200 mg of a medicine has a half-life of 2 hours. Write the amount $A(t)$ , in mg, left after $t$ hours.
A sample of 200 mg has a half-life of 2 years. How much is left after 8 years? Round to 2 decimal places.
Evaluate $\log_{ 10 } 20 + \log_{ 10 } 5$ without a calculator.
Evaluate $2\log_{ 2 } 6 - \log_{ 2 } 9$ without a calculator.
Which expression is equal to $\log_{ 8 }(2 \cdot 7)$ ?
1. $\log_{ 8 } 2 + \log_{ 8 } 7$
2. $\log_{ 8 } 2 \cdot \log_{ 8 } 7$
3. $\log_{ 8 }(2 + 7)$
4. $7 \log_{ 8 } 2$
Which expression is equal to $\log_{ 7 }(4 \cdot 9)$ ?
1. $\log_{ 7 } 4 + \log_{ 7 } 9$
2. $\log_{ 7 } 4 \cdot \log_{ 7 } 9$
3. $\log_{ 7 }(4 + 9)$
4. $9 \log_{ 7 } 4$
Which expression is equal to $\log_{ 7 }(5 \cdot 3)$ ?
1. $\log_{ 7 } 5 + \log_{ 7 } 3$
2. $\log_{ 7 } 5 \cdot \log_{ 7 } 3$
3. $\log_{ 7 }(5 + 3)$
4. $3 \log_{ 7 } 5$
Which expression is equal to $\log_{ 7 }(5 \cdot 8)$ ?
1. $\log_{ 7 } 5 + \log_{ 7 } 8$
2. $\log_{ 7 } 5 \cdot \log_{ 7 } 8$
3. $\log_{ 7 }(5 + 8)$
4. $8 \log_{ 7 } 5$
Which expression is equal to $\log_{ 7 }(8 \cdot 9)$ ?
1. $\log_{ 7 } 8 + \log_{ 7 } 9$
2. $\log_{ 7 } 8 \cdot \log_{ 7 } 9$
3. $\log_{ 7 }(8 + 9)$
4. $9 \log_{ 7 } 8$
Which expression is equal to $\log_{ 8 }(3 \cdot 6)$ ?
1. $\log_{ 8 } 3 + \log_{ 8 } 6$
2. $\log_{ 8 } 3 \cdot \log_{ 8 } 6$
3. $\log_{ 8 }(3 + 6)$
4. $6 \log_{ 8 } 3$
Which expression is equal to $\log_{ 5 }(3 \cdot 4)$ ?
1. $\log_{ 5 } 3 + \log_{ 5 } 4$
2. $\log_{ 5 } 3 \cdot \log_{ 5 } 4$
3. $\log_{ 5 }(3 + 4)$
4. $4 \log_{ 5 } 3$
Which expression is equal to $\log_{ 5 }(2 \cdot 9)$ ?
1. $\log_{ 5 } 2 + \log_{ 5 } 9$
2. $\log_{ 5 } 2 \cdot \log_{ 5 } 9$
3. $\log_{ 5 }(2 + 9)$
4. $9 \log_{ 5 } 2$
Which expression is equal to $\log_{ 3 }(7 \cdot 4)$ ?
1. $\log_{ 3 } 7 + \log_{ 3 } 4$
2. $\log_{ 3 } 7 \cdot \log_{ 3 } 4$
3. $\log_{ 3 }(7 + 4)$
4. $4 \log_{ 3 } 7$
Which expression is equal to $\log_{ 6 }(7 \cdot 8)$ ?
1. $\log_{ 6 } 7 + \log_{ 6 } 8$
2. $\log_{ 6 } 7 \cdot \log_{ 6 } 8$
3. $\log_{ 6 }(7 + 8)$
4. $8 \log_{ 6 } 7$
Which expression is equal to $\log_{ 3 }(5 \cdot 6)$ ?
1. $\log_{ 3 } 5 + \log_{ 3 } 6$
2. $\log_{ 3 } 5 \cdot \log_{ 3 } 6$
3. $\log_{ 3 }(5 + 6)$
4. $6 \log_{ 3 } 5$
Which expression is equal to $\log_{ 9 }(7 \cdot 5)$ ?
1. $\log_{ 9 } 7 + \log_{ 9 } 5$
2. $\log_{ 9 } 7 \cdot \log_{ 9 } 5$
3. $\log_{ 9 }(7 + 5)$
4. $5 \log_{ 9 } 7$
Which expression is equal to $\log_{ 7 }(2 \cdot 5)$ ?
1. $\log_{ 7 } 2 + \log_{ 7 } 5$
2. $\log_{ 7 } 2 \cdot \log_{ 7 } 5$
3. $\log_{ 7 }(2 + 5)$
4. $5 \log_{ 7 } 2$
Which expression is equal to $\log_{ 6 }(2 \cdot 7)$ ?
1. $\log_{ 6 } 2 + \log_{ 6 } 7$
2. $\log_{ 6 } 2 \cdot \log_{ 6 } 7$
3. $\log_{ 6 }(2 + 7)$
4. $7 \log_{ 6 } 2$
Which expression is equal to $\log_{ 4 }(5 \cdot 9)$ ?
1. $\log_{ 4 } 5 + \log_{ 4 } 9$
2. $\log_{ 4 } 5 \cdot \log_{ 4 } 9$
3. $\log_{ 4 }(5 + 9)$
4. $9 \log_{ 4 } 5$
Which expression is equal to $\log_{ 3 }(5 \cdot 2)$ ?
1. $\log_{ 3 } 5 + \log_{ 3 } 2$
2. $\log_{ 3 } 5 \cdot \log_{ 3 } 2$
3. $\log_{ 3 }(5 + 2)$
4. $2 \log_{ 3 } 5$
Which expression is equal to $\log_{ 3 }(2 \cdot 6)$ ?
1. $\log_{ 3 } 2 + \log_{ 3 } 6$
2. $\log_{ 3 } 2 \cdot \log_{ 3 } 6$
3. $\log_{ 3 }(2 + 6)$
4. $6 \log_{ 3 } 2$
Which expression is equal to $\log_{ 4 }(9 \cdot 7)$ ?
1. $\log_{ 4 } 9 + \log_{ 4 } 7$
2. $\log_{ 4 } 9 \cdot \log_{ 4 } 7$
3. $\log_{ 4 }(9 + 7)$
4. $7 \log_{ 4 } 9$
Which expression is equal to $\log_{ 7 }(4 \cdot 3)$ ?
1. $\log_{ 7 } 4 + \log_{ 7 } 3$
2. $\log_{ 7 } 4 \cdot \log_{ 7 } 3$
3. $\log_{ 7 }(4 + 3)$
4. $3 \log_{ 7 } 4$
Which expression is equal to $\log_{ 8 }(5 \cdot 6)$ ?
1. $\log_{ 8 } 5 + \log_{ 8 } 6$
2. $\log_{ 8 } 5 \cdot \log_{ 8 } 6$
3. $\log_{ 8 }(5 + 6)$
4. $6 \log_{ 8 } 5$
Which expression is equal to $\log_{ 3 }(2 \cdot 7)$ ?
1. $\log_{ 3 } 2 + \log_{ 3 } 7$
2. $\log_{ 3 } 2 \cdot \log_{ 3 } 7$
3. $\log_{ 3 }(2 + 7)$
4. $7 \log_{ 3 } 2$
Which expression is equal to $\log_{ 2 }(5 \cdot 3)$ ?
1. $\log_{ 2 } 5 + \log_{ 2 } 3$
2. $\log_{ 2 } 5 \cdot \log_{ 2 } 3$
3. $\log_{ 2 }(5 + 3)$
4. $3 \log_{ 2 } 5$
Which expression is equal to $\log_{ 9 }(7 \cdot 8)$ ?
1. $\log_{ 9 } 7 + \log_{ 9 } 8$
2. $\log_{ 9 } 7 \cdot \log_{ 9 } 8$
3. $\log_{ 9 }(7 + 8)$
4. $8 \log_{ 9 } 7$
Which expression is equal to $\log_{ 7 }(4 \cdot 8)$ ?
1. $\log_{ 7 } 4 + \log_{ 7 } 8$
2. $\log_{ 7 } 4 \cdot \log_{ 7 } 8$
3. $\log_{ 7 }(4 + 8)$
4. $8 \log_{ 7 } 4$
Which expression is equal to $\log_{ 2 }(7 \cdot 6)$ ?
1. $\log_{ 2 } 7 + \log_{ 2 } 6$
2. $\log_{ 2 } 7 \cdot \log_{ 2 } 6$
3. $\log_{ 2 }(7 + 6)$
4. $6 \log_{ 2 } 7$
Which expression is equal to $\log_{ 8 }(2 \cdot 4)$ ?
1. $\log_{ 8 } 2 + \log_{ 8 } 4$
2. $\log_{ 8 } 2 \cdot \log_{ 8 } 4$
3. $\log_{ 8 }(2 + 4)$
4. $4 \log_{ 8 } 2$
Which expression is equal to $\log_{ 9 }(4 \cdot 7)$ ?
1. $\log_{ 9 } 4 + \log_{ 9 } 7$
2. $\log_{ 9 } 4 \cdot \log_{ 9 } 7$
3. $\log_{ 9 }(4 + 7)$
4. $7 \log_{ 9 } 4$
Which expression is equal to $\log_{ 7 }(6 \cdot 9)$ ?
1. $\log_{ 7 } 6 + \log_{ 7 } 9$
2. $\log_{ 7 } 6 \cdot \log_{ 7 } 9$
3. $\log_{ 7 }(6 + 9)$
4. $9 \log_{ 7 } 6$
Which expression is equal to $\log_{ 6 }(7 \cdot 9)$ ?
1. $\log_{ 6 } 7 + \log_{ 6 } 9$
2. $\log_{ 6 } 7 \cdot \log_{ 6 } 9$
3. $\log_{ 6 }(7 + 9)$
4. $9 \log_{ 6 } 7$
Which expression is equal to $\log_{ 9 }(6 \cdot 4)$ ?
1. $\log_{ 9 } 6 + \log_{ 9 } 4$
2. $\log_{ 9 } 6 \cdot \log_{ 9 } 4$
3. $\log_{ 9 }(6 + 4)$
4. $4 \log_{ 9 } 6$
Which expression is equal to $\log_{ 9 }(3 \cdot 2)$ ?
1. $\log_{ 9 } 3 + \log_{ 9 } 2$
2. $\log_{ 9 } 3 \cdot \log_{ 9 } 2$
3. $\log_{ 9 }(3 + 2)$
4. $2 \log_{ 9 } 3$
Which expression is equal to $\log_{ 3 }(8 \cdot 6)$ ?
1. $\log_{ 3 } 8 + \log_{ 3 } 6$
2. $\log_{ 3 } 8 \cdot \log_{ 3 } 6$
3. $\log_{ 3 }(8 + 6)$
4. $6 \log_{ 3 } 8$
Which expression is equal to $\log_{ 4 }(8 \cdot 5)$ ?
1. $\log_{ 4 } 8 + \log_{ 4 } 5$
2. $\log_{ 4 } 8 \cdot \log_{ 4 } 5$
3. $\log_{ 4 }(8 + 5)$
4. $5 \log_{ 4 } 8$
Which expression is equal to $\log_{ 2 }(4 \cdot 6)$ ?
1. $\log_{ 2 } 4 + \log_{ 2 } 6$
2. $\log_{ 2 } 4 \cdot \log_{ 2 } 6$
3. $\log_{ 2 }(4 + 6)$
4. $6 \log_{ 2 } 4$
Which expression is equal to $\log_{ 3 }(9 \cdot 5)$ ?
1. $\log_{ 3 } 9 + \log_{ 3 } 5$
2. $\log_{ 3 } 9 \cdot \log_{ 3 } 5$
3. $\log_{ 3 }(9 + 5)$
4. $5 \log_{ 3 } 9$
Which expression is equal to $\log_{ 3 }(8 \cdot 7)$ ?
1. $\log_{ 3 } 8 + \log_{ 3 } 7$
2. $\log_{ 3 } 8 \cdot \log_{ 3 } 7$
3. $\log_{ 3 }(8 + 7)$
4. $7 \log_{ 3 } 8$
Which expression is equal to $\log_{ 2 }(6 \cdot 9)$ ?
1. $\log_{ 2 } 6 + \log_{ 2 } 9$
2. $\log_{ 2 } 6 \cdot \log_{ 2 } 9$
3. $\log_{ 2 }(6 + 9)$
4. $9 \log_{ 2 } 6$
Which expression is equal to $\log_{ 3 }(7 \cdot 5)$ ?
1. $\log_{ 3 } 7 + \log_{ 3 } 5$
2. $\log_{ 3 } 7 \cdot \log_{ 3 } 5$
3. $\log_{ 3 }(7 + 5)$
4. $5 \log_{ 3 } 7$
Which expression is equal to $\log_{ 9 }(8 \cdot 7)$ ?
1. $\log_{ 9 } 8 + \log_{ 9 } 7$
2. $\log_{ 9 } 8 \cdot \log_{ 9 } 7$
3. $\log_{ 9 }(8 + 7)$
4. $7 \log_{ 9 } 8$
Which expression is equal to $\log_{ 8 }(6 \cdot 7)$ ?
1. $\log_{ 8 } 6 + \log_{ 8 } 7$
2. $\log_{ 8 } 6 \cdot \log_{ 8 } 7$
3. $\log_{ 8 }(6 + 7)$
4. $7 \log_{ 8 } 6$
Which expression is equal to $\log_{ 2 }(3 \cdot 5)$ ?
1. $\log_{ 2 } 3 + \log_{ 2 } 5$
2. $\log_{ 2 } 3 \cdot \log_{ 2 } 5$
3. $\log_{ 2 }(3 + 5)$
4. $5 \log_{ 2 } 3$
Which expression is equal to $\log_{ 6 }(3 \cdot 7)$ ?
1. $\log_{ 6 } 3 + \log_{ 6 } 7$
2. $\log_{ 6 } 3 \cdot \log_{ 6 } 7$
3. $\log_{ 6 }(3 + 7)$
4. $7 \log_{ 6 } 3$
Which expression is equal to $\log_{ 6 }(2 \cdot 5)$ ?
1. $\log_{ 6 } 2 + \log_{ 6 } 5$
2. $\log_{ 6 } 2 \cdot \log_{ 6 } 5$
3. $\log_{ 6 }(2 + 5)$
4. $5 \log_{ 6 } 2$
Which expression is equal to $\log_{ 9 }(8 \cdot 4)$ ?
1. $\log_{ 9 } 8 + \log_{ 9 } 4$
2. $\log_{ 9 } 8 \cdot \log_{ 9 } 4$
3. $\log_{ 9 }(8 + 4)$
4. $4 \log_{ 9 } 8$
Which expression is equal to $\log_{ 3 }(2 \cdot 8)$ ?
1. $\log_{ 3 } 2 + \log_{ 3 } 8$
2. $\log_{ 3 } 2 \cdot \log_{ 3 } 8$
3. $\log_{ 3 }(2 + 8)$
4. $8 \log_{ 3 } 2$
Which expression is equal to $\log_{ 7 }(3 \cdot 9)$ ?
1. $\log_{ 7 } 3 + \log_{ 7 } 9$
2. $\log_{ 7 } 3 \cdot \log_{ 7 } 9$
3. $\log_{ 7 }(3 + 9)$
4. $9 \log_{ 7 } 3$
Which expression is equal to $\log_{ 7 }(3 \cdot 2)$ ?
1. $\log_{ 7 } 3 + \log_{ 7 } 2$
2. $\log_{ 7 } 3 \cdot \log_{ 7 } 2$
3. $\log_{ 7 }(3 + 2)$
4. $2 \log_{ 7 } 3$
Which expression is equal to $\log_{ 8 }(9 \cdot 2)$ ?
1. $\log_{ 8 } 9 + \log_{ 8 } 2$
2. $\log_{ 8 } 9 \cdot \log_{ 8 } 2$
3. $\log_{ 8 }(9 + 2)$
4. $2 \log_{ 8 } 9$
Which expression is equal to $\log_{ 5 }(3 \cdot 7)$ ?
1. $\log_{ 5 } 3 + \log_{ 5 } 7$
2. $\log_{ 5 } 3 \cdot \log_{ 5 } 7$
3. $\log_{ 5 }(3 + 7)$
4. $7 \log_{ 5 } 3$
Which expression is equal to $\log_{ 4 }(2 \cdot 8)$ ?
1. $\log_{ 4 } 2 + \log_{ 4 } 8$
2. $\log_{ 4 } 2 \cdot \log_{ 4 } 8$
3. $\log_{ 4 }(2 + 8)$
4. $8 \log_{ 4 } 2$
Write $\log_{ 3 } x + 2\log_{ 3 } 4$ as a single logarithm.
Expand $\log_{ 8 }\left(\dfrac{x^{ 4 }}{y^{ 5 }}\right)$ : write it with logarithms of $x$ and $y$ only.
Complete the change of base: $\log_{ 4 } 20 = \dfrac{\ln 20 }{\square}$ . What goes in the box?
Evaluate $\log_{ 8 } 16$ exactly. Hint: write 8 and 16 as powers of the same number.
Write $\log_{ 8 } 16$ as a fraction in lowest terms.
Place $\log_{2} 6$ on the number line.
Solve $e^{ 5x } = 25$ .
Solve $e^{ 4x + 4 } = 12$ .
Solve $3e^{ 4x } + 6 = 60$ .
Solve $6^x = 35$ . Give the exact answer using natural logarithms.
Solve $2\cdot 3^x = 48$ . Give the exact answer.
Solve $\log_{ 5 }(x - 8) = 4$ .
Solve $\ln x + \ln(x - 2) = \ln 3$ .
Solve $\ln x + \ln(x - 1) = \ln 42$ . Give every solution.
Put the steps for solving $\ln x + \ln(x - 5) = \ln 6$ in order.
- Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 6$
- Equal logarithms have equal arguments: $x(x - 5) = 6$
- Solve the quadratic equation
- Keep only the values where $x > 5$
Put the steps for solving $\ln x + \ln(x - 4) = \ln 21$ in order.
- Combine with the product rule: $\ln\big(x(x - 4)\big) = \ln 21$
- Equal logarithms have equal arguments: $x(x - 4) = 21$
- Solve the quadratic equation
- Keep only the values where $x > 4$
Put the steps for solving $\ln x + \ln(x - 3) = \ln 7$ in order.
- Combine with the product rule: $\ln\big(x(x - 3)\big) = \ln 7$
- Equal logarithms have equal arguments: $x(x - 3) = 7$
- Solve the quadratic equation
- Keep only the values where $x > 3$
Put the steps for solving $\ln x + \ln(x - 2) = \ln 23$ in order.
- Combine with the product rule: $\ln\big(x(x - 2)\big) = \ln 23$
- Equal logarithms have equal arguments: $x(x - 2) = 23$
- Solve the quadratic equation
- Keep only the values where $x > 2$
Put the steps for solving $\ln x + \ln(x - 2) = \ln 38$ in order.
- Combine with the product rule: $\ln\big(x(x - 2)\big) = \ln 38$
- Equal logarithms have equal arguments: $x(x - 2) = 38$
- Solve the quadratic equation
- Keep only the values where $x > 2$
Put the steps for solving $\ln x + \ln(x - 6) = \ln 26$ in order.
- Combine with the product rule: $\ln\big(x(x - 6)\big) = \ln 26$
- Equal logarithms have equal arguments: $x(x - 6) = 26$
- Solve the quadratic equation
- Keep only the values where $x > 6$
Put the steps for solving $\ln x + \ln(x - 5) = \ln 38$ in order.
- Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 38$
- Equal logarithms have equal arguments: $x(x - 5) = 38$
- Solve the quadratic equation
- Keep only the values where $x > 5$
Put the steps for solving $\ln x + \ln(x - 5) = \ln 34$ in order.
- Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 34$
- Equal logarithms have equal arguments: $x(x - 5) = 34$
- Solve the quadratic equation
- Keep only the values where $x > 5$
Put the steps for solving $\ln x + \ln(x - 4) = \ln 33$ in order.
- Combine with the product rule: $\ln\big(x(x - 4)\big) = \ln 33$
- Equal logarithms have equal arguments: $x(x - 4) = 33$
- Solve the quadratic equation
- Keep only the values where $x > 4$
Put the steps for solving $\ln x + \ln(x - 6) = \ln 8$ in order.
- Combine with the product rule: $\ln\big(x(x - 6)\big) = \ln 8$
- Equal logarithms have equal arguments: $x(x - 6) = 8$
- Solve the quadratic equation
- Keep only the values where $x > 6$
Put the steps for solving $\ln x + \ln(x - 1) = \ln 30$ in order.
- Combine with the product rule: $\ln\big(x(x - 1)\big) = \ln 30$
- Equal logarithms have equal arguments: $x(x - 1) = 30$
- Solve the quadratic equation
- Keep only the values where $x > 1$
Put the steps for solving $\ln x + \ln(x - 3) = \ln 37$ in order.
- Combine with the product rule: $\ln\big(x(x - 3)\big) = \ln 37$
- Equal logarithms have equal arguments: $x(x - 3) = 37$
- Solve the quadratic equation
- Keep only the values where $x > 3$
Put the steps for solving $\ln x + \ln(x - 1) = \ln 35$ in order.
- Combine with the product rule: $\ln\big(x(x - 1)\big) = \ln 35$
- Equal logarithms have equal arguments: $x(x - 1) = 35$
- Solve the quadratic equation
- Keep only the values where $x > 1$
Put the steps for solving $\ln x + \ln(x - 4) = \ln 27$ in order.
- Combine with the product rule: $\ln\big(x(x - 4)\big) = \ln 27$
- Equal logarithms have equal arguments: $x(x - 4) = 27$
- Solve the quadratic equation
- Keep only the values where $x > 4$
Put the steps for solving $\ln x + \ln(x - 3) = \ln 25$ in order.
- Combine with the product rule: $\ln\big(x(x - 3)\big) = \ln 25$
- Equal logarithms have equal arguments: $x(x - 3) = 25$
- Solve the quadratic equation
- Keep only the values where $x > 3$
Put the steps for solving $\ln x + \ln(x - 1) = \ln 24$ in order.
- Combine with the product rule: $\ln\big(x(x - 1)\big) = \ln 24$
- Equal logarithms have equal arguments: $x(x - 1) = 24$
- Solve the quadratic equation
- Keep only the values where $x > 1$
Put the steps for solving $\ln x + \ln(x - 1) = \ln 22$ in order.
- Combine with the product rule: $\ln\big(x(x - 1)\big) = \ln 22$
- Equal logarithms have equal arguments: $x(x - 1) = 22$
- Solve the quadratic equation
- Keep only the values where $x > 1$
Put the steps for solving $\ln x + \ln(x - 2) = \ln 6$ in order.
- Combine with the product rule: $\ln\big(x(x - 2)\big) = \ln 6$
- Equal logarithms have equal arguments: $x(x - 2) = 6$
- Solve the quadratic equation
- Keep only the values where $x > 2$
Put the steps for solving $\ln x + \ln(x - 6) = \ln 30$ in order.
- Combine with the product rule: $\ln\big(x(x - 6)\big) = \ln 30$
- Equal logarithms have equal arguments: $x(x - 6) = 30$
- Solve the quadratic equation
- Keep only the values where $x > 6$
Put the steps for solving $\ln x + \ln(x - 4) = \ln 6$ in order.
- Combine with the product rule: $\ln\big(x(x - 4)\big) = \ln 6$
- Equal logarithms have equal arguments: $x(x - 4) = 6$
- Solve the quadratic equation
- Keep only the values where $x > 4$
Put the steps for solving $\ln x + \ln(x - 3) = \ln 24$ in order.
- Combine with the product rule: $\ln\big(x(x - 3)\big) = \ln 24$
- Equal logarithms have equal arguments: $x(x - 3) = 24$
- Solve the quadratic equation
- Keep only the values where $x > 3$
Put the steps for solving $\ln x + \ln(x - 2) = \ln 10$ in order.
- Combine with the product rule: $\ln\big(x(x - 2)\big) = \ln 10$
- Equal logarithms have equal arguments: $x(x - 2) = 10$
- Solve the quadratic equation
- Keep only the values where $x > 2$
Put the steps for solving $\ln x + \ln(x - 3) = \ln 34$ in order.
- Combine with the product rule: $\ln\big(x(x - 3)\big) = \ln 34$
- Equal logarithms have equal arguments: $x(x - 3) = 34$
- Solve the quadratic equation
- Keep only the values where $x > 3$
Put the steps for solving $\ln x + \ln(x - 6) = \ln 16$ in order.
- Combine with the product rule: $\ln\big(x(x - 6)\big) = \ln 16$
- Equal logarithms have equal arguments: $x(x - 6) = 16$
- Solve the quadratic equation
- Keep only the values where $x > 6$
Put the steps for solving $\ln x + \ln(x - 2) = \ln 12$ in order.
- Combine with the product rule: $\ln\big(x(x - 2)\big) = \ln 12$
- Equal logarithms have equal arguments: $x(x - 2) = 12$
- Solve the quadratic equation
- Keep only the values where $x > 2$
Put the steps for solving $\ln x + \ln(x - 5) = \ln 15$ in order.
- Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 15$
- Equal logarithms have equal arguments: $x(x - 5) = 15$
- Solve the quadratic equation
- Keep only the values where $x > 5$
Put the steps for solving $\ln x + \ln(x - 6) = \ln 9$ in order.
- Combine with the product rule: $\ln\big(x(x - 6)\big) = \ln 9$
- Equal logarithms have equal arguments: $x(x - 6) = 9$
- Solve the quadratic equation
- Keep only the values where $x > 6$
Put the steps for solving $\ln x + \ln(x - 6) = \ln 37$ in order.
- Combine with the product rule: $\ln\big(x(x - 6)\big) = \ln 37$
- Equal logarithms have equal arguments: $x(x - 6) = 37$
- Solve the quadratic equation
- Keep only the values where $x > 6$
Put the steps for solving $\ln x + \ln(x - 6) = \ln 31$ in order.
- Combine with the product rule: $\ln\big(x(x - 6)\big) = \ln 31$
- Equal logarithms have equal arguments: $x(x - 6) = 31$
- Solve the quadratic equation
- Keep only the values where $x > 6$
Put the steps for solving $\ln x + \ln(x - 2) = \ln 21$ in order.
- Combine with the product rule: $\ln\big(x(x - 2)\big) = \ln 21$
- Equal logarithms have equal arguments: $x(x - 2) = 21$
- Solve the quadratic equation
- Keep only the values where $x > 2$
Put the steps for solving $\ln x + \ln(x - 4) = \ln 20$ in order.
- Combine with the product rule: $\ln\big(x(x - 4)\big) = \ln 20$
- Equal logarithms have equal arguments: $x(x - 4) = 20$
- Solve the quadratic equation
- Keep only the values where $x > 4$
Put the steps for solving $\ln x + \ln(x - 2) = \ln 18$ in order.
- Combine with the product rule: $\ln\big(x(x - 2)\big) = \ln 18$
- Equal logarithms have equal arguments: $x(x - 2) = 18$
- Solve the quadratic equation
- Keep only the values where $x > 2$
Put the steps for solving $\ln x + \ln(x - 5) = \ln 8$ in order.
- Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 8$
- Equal logarithms have equal arguments: $x(x - 5) = 8$
- Solve the quadratic equation
- Keep only the values where $x > 5$
Put the steps for solving $\ln x + \ln(x - 1) = \ln 23$ in order.
- Combine with the product rule: $\ln\big(x(x - 1)\big) = \ln 23$
- Equal logarithms have equal arguments: $x(x - 1) = 23$
- Solve the quadratic equation
- Keep only the values where $x > 1$
Put the steps for solving $\ln x + \ln(x - 5) = \ln 12$ in order.
- Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 12$
- Equal logarithms have equal arguments: $x(x - 5) = 12$
- Solve the quadratic equation
- Keep only the values where $x > 5$
Put the steps for solving $\ln x + \ln(x - 3) = \ln 23$ in order.
- Combine with the product rule: $\ln\big(x(x - 3)\big) = \ln 23$
- Equal logarithms have equal arguments: $x(x - 3) = 23$
- Solve the quadratic equation
- Keep only the values where $x > 3$
Put the steps for solving $\ln x + \ln(x - 1) = \ln 29$ in order.
- Combine with the product rule: $\ln\big(x(x - 1)\big) = \ln 29$
- Equal logarithms have equal arguments: $x(x - 1) = 29$
- Solve the quadratic equation
- Keep only the values where $x > 1$
Put the steps for solving $\ln x + \ln(x - 6) = \ln 36$ in order.
- Combine with the product rule: $\ln\big(x(x - 6)\big) = \ln 36$
- Equal logarithms have equal arguments: $x(x - 6) = 36$
- Solve the quadratic equation
- Keep only the values where $x > 6$
Put the steps for solving $\ln x + \ln(x - 5) = \ln 28$ in order.
- Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 28$
- Equal logarithms have equal arguments: $x(x - 5) = 28$
- Solve the quadratic equation
- Keep only the values where $x > 5$
Put the steps for solving $\ln x + \ln(x - 5) = \ln 31$ in order.
- Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 31$
- Equal logarithms have equal arguments: $x(x - 5) = 31$
- Solve the quadratic equation
- Keep only the values where $x > 5$
Put the steps for solving $\ln x + \ln(x - 5) = \ln 23$ in order.
- Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 23$
- Equal logarithms have equal arguments: $x(x - 5) = 23$
- Solve the quadratic equation
- Keep only the values where $x > 5$
Put the steps for solving $\ln x + \ln(x - 6) = \ln 20$ in order.
- Combine with the product rule: $\ln\big(x(x - 6)\big) = \ln 20$
- Equal logarithms have equal arguments: $x(x - 6) = 20$
- Solve the quadratic equation
- Keep only the values where $x > 6$
Put the steps for solving $\ln x + \ln(x - 3) = \ln 17$ in order.
- Combine with the product rule: $\ln\big(x(x - 3)\big) = \ln 17$
- Equal logarithms have equal arguments: $x(x - 3) = 17$
- Solve the quadratic equation
- Keep only the values where $x > 3$
Put the steps for solving $\ln x + \ln(x - 3) = \ln 26$ in order.
- Combine with the product rule: $\ln\big(x(x - 3)\big) = \ln 26$
- Equal logarithms have equal arguments: $x(x - 3) = 26$
- Solve the quadratic equation
- Keep only the values where $x > 3$
Put the steps for solving $\ln x + \ln(x - 2) = \ln 40$ in order.
- Combine with the product rule: $\ln\big(x(x - 2)\big) = \ln 40$
- Equal logarithms have equal arguments: $x(x - 2) = 40$
- Solve the quadratic equation
- Keep only the values where $x > 2$
Put the steps for solving $\ln x + \ln(x - 4) = \ln 9$ in order.
- Combine with the product rule: $\ln\big(x(x - 4)\big) = \ln 9$
- Equal logarithms have equal arguments: $x(x - 4) = 9$
- Solve the quadratic equation
- Keep only the values where $x > 4$
Put the steps for solving $\ln x + \ln(x - 4) = \ln 37$ in order.
- Combine with the product rule: $\ln\big(x(x - 4)\big) = \ln 37$
- Equal logarithms have equal arguments: $x(x - 4) = 37$
- Solve the quadratic equation
- Keep only the values where $x > 4$
Put the steps for solving $\ln x + \ln(x - 3) = \ln 28$ in order.
- Combine with the product rule: $\ln\big(x(x - 3)\big) = \ln 28$
- Equal logarithms have equal arguments: $x(x - 3) = 28$
- Solve the quadratic equation
- Keep only the values where $x > 3$
Put the steps for solving $\ln x + \ln(x - 5) = \ln 14$ in order.
- Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 14$
- Equal logarithms have equal arguments: $x(x - 5) = 14$
- Solve the quadratic equation
- Keep only the values where $x > 5$
Put the steps for solving $\ln x + \ln(x - 6) = \ln 12$ in order.
- Combine with the product rule: $\ln\big(x(x - 6)\big) = \ln 12$
- Equal logarithms have equal arguments: $x(x - 6) = 12$
- Solve the quadratic equation
- Keep only the values where $x > 6$

Answers

$2$
$\frac{\ln{\left(21 \right)}}{\ln{\left(6 \right)}}$
$\log_{ 8 } y = 2$ → $y = 8^{2}$ · $\log_{ 8 } y = 3$ → $y = 8^{3}$ · $\log_{ 8 } y = -1$ → $y = \dfrac{1}{ 8 }$ · $\log_{ 8 } y = \tfrac{1}{2}$ → $y = \sqrt{ 8 }$
$7$
$6$
$y = 2^{x} + 3$
$8$
$200 \cdot 2^{- \frac{t}{2}}$
$\frac{25}{2}$ mg
$2$
$2$
$\log_{ 8 } 2 + \log_{ 8 } 7$
$\frac{\ln{\left(16 x \right)}}{\ln{\left(3 \right)}}$
$\frac{4 \ln{\left(x \right)}}{\ln{\left(8 \right)}} - \frac{5 \ln{\left(y \right)}}{\ln{\left(8 \right)}}$
$\ln{\left(4 \right)}$
$\frac{4}{3}$
$\frac{4}{3}$
$1 + \frac{\ln{\left(3 \right)}}{\ln{\left(2 \right)}}$
$\frac{\ln{\left(25 \right)}}{5}$
$-1 + \frac{\ln{\left(12 \right)}}{4}$
$\frac{\ln{\left(18 \right)}}{4}$
$\frac{\ln{\left(35 \right)}}{\ln{\left(6 \right)}}$
$\frac{\ln{\left(24 \right)}}{\ln{\left(3 \right)}}$
$633$
$3$
$7$
1. Combine with the product rule: $\ln\big(x(x - 5)\big) = \ln 6$ 2. Equal logarithms have equal arguments: $x(x - 5) = 6$ 3. Solve the quadratic equation 4. Keep only the values where $x > 5$