Exponential and logarithmic models

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1. Interpret exponential functions and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

f(x)=40⋅(12)xf(x) = 40 \cdot \left(\tfrac{1}{2}\right)^x. What is f(0)f(0)?

  1. 4040
  2. 00

Key idea

Read aa and bb from f(x)=a⋅bxf(x) = a \cdot b^x.

a=f(0)a = f(0) is the starting value (the yy-intercept). bb is the growth factor: b>1b > 1 growth, 0<b<10 < b < 1 decay. Negative inputs go back in time: f(−1)=a÷bf(-1) = a \div b.

Worked example

Graph through (0,3)(0, 3) and (1,12)(1, 12).

a=3a = 3
Starting value.
b=4b = 4
12 ÷ 3.
f(−1)=3÷4=0.75f(-1) = 3 \div 4 = 0.75
One step back.

Your turn, guided

Your turn: f(x)=50⋅5xf(x) = 50 \cdot 5^x. Find f(−1)f(-1).

1010
50 ÷ 5.

Practice ★

  1. f(x)=24⋅4xf(x) = 24 \cdot 4^x. Find f(−1)f(-1).
  2. f(x)=18⋅3xf(x) = 18 \cdot 3^x. Find f(−1)f(-1).
  1. Is f(x)=4⋅(14)xf(x) = 4 \cdot \left(\tfrac{1}{ 4 }\right)^x exponential growth or decay?
    1. Decay: each step multiplies by 14\tfrac{1}{ 4 }
    2. Growth, because 4 is greater than 1
    3. A linear decrease

Watch out!

Practice ★★

  1. The graph of an exponential function passes through (0,8)(0, 8) and (1,16)(1, 16). Write f(x)f(x).
  2. The graph of an exponential function passes through (0,7)(0, 7) and (1,21)(1, 21). Write f(x)f(x).

Exit ticket

Three questions to check you have got it.

  1. f(x)=24⋅4xf(x) = 24 \cdot 4^x. Find f(−1)f(-1).
  2. f(x)=18⋅3xf(x) = 18 \cdot 3^x. Find f(−1)f(-1).
  3. f(x)=27⋅3xf(x) = 27 \cdot 3^x. Find f(−1)f(-1).

Summary card

Reading exponential functions

aa: value at x=0x = 0; bb: factor per step; b−1=1/bb^{-1} = 1/b.

Watch out: negative exponent, positive value.

2. What is a logarithm? Logarithms as exponents

Warm-up

Three quick questions from earlier lessons.

Discover

Here is part of a table of powers of 10: 100=110^0 = 1, 101=1010^1 = 10, 102=10010^2 = 100, 103=100010^3 = 1000.

Solve 10x=100010^x = 1000. Now try 10x=5010^x = 50: between which two whole numbers is xx?

Key idea

A logarithm is an exponent. log⁡bx\log_b x answers the question: "bb to what power gives xx?"

log⁡bx=y  ⟺  by=x\log_b x = y \iff b^y = x (with b>0b > 0, b≠1b \ne 1, x>0x > 0)

Written without a base, log⁡x\log x means log⁡10x\log_{10} x. So the solution of 10x=5010^x = 50 is x=log⁡50≈1.7x = \log 50 \approx 1.7.

Worked example

Evaluate log⁡232\log_2 32.

2y=322^{y} = 32
Ask: 2 to what power gives 32?
y=5y = 5
2⁵ = 32.
log⁡232=5\log_2 32 = 5
So the logarithm is 5.

Your turn, guided

Your turn: evaluate log⁡381\log_3 81. Fill in the missing line.

3y=813^{y} = 81
Ask: 3 to what power gives 81?
log⁡381=4\log_3 81 = 4
3⁴ = 81.

Practice ★

  1. Evaluate log⁡416\log_{ 4 } 16.
  2. Evaluate log⁡525\log_{ 5 } 25.
  3. Evaluate log⁡4256\log_{ 4 } 256.
  4. Evaluate log⁡464\log_{ 4 } 64.
  5. Evaluate log⁡28\log_{ 2 } 8.

Watch out!

Practice ★★

  1. Match each logarithm statement with the same statement written as a power.
    1. log⁡8y=2\log_{ 8 } y = 2
    2. log⁡8y=3\log_{ 8 } y = 3
    3. log⁡8y=−1\log_{ 8 } y = -1
    4. log⁡8y=12\log_{ 8 } y = \tfrac{1}{2}
    1. y=82y = 8^{2}
    2. y=83y = 8^{3}
    3. y=18y = \dfrac{1}{ 8 }
    4. y=8y = \sqrt{ 8 }
  2. Match each logarithm statement with the same statement written as a power.
    1. log⁡3y=2\log_{ 3 } y = 2
    2. log⁡3y=3\log_{ 3 } y = 3
    3. log⁡3y=−1\log_{ 3 } y = -1
    4. log⁡3y=12\log_{ 3 } y = \tfrac{1}{2}
    1. y=32y = 3^{2}
    2. y=33y = 3^{3}
    3. y=13y = \dfrac{1}{ 3 }
    4. y=3y = \sqrt{ 3 }
  1. Solve 6x=216^x = 21. Write the exact answer using a logarithm.
  2. Solve 7x=167^x = 16. Write the exact answer using a logarithm.
  3. Solve 2x=312^x = 31. Write the exact answer using a logarithm.
  4. Solve 7x=157^x = 15. Write the exact answer using a logarithm.
  5. Solve 2x=372^x = 37. Write the exact answer using a logarithm.

Exit ticket

Three questions to check you have got it.

  1. Solve 6x=216^x = 21. Write the exact answer using a logarithm.
  2. Solve 7x=167^x = 16. Write the exact answer using a logarithm.
  3. Solve 2x=312^x = 31. Write the exact answer using a logarithm.

Summary card

Logarithms are exponents

  • log⁡bx=y\log_b x = y means by=xb^y = x.
  • log⁡x\log x means log⁡10x\log_{10} x.
  • The solution of bx=cb^x = c is x=log⁡bcx = \log_b c.

Watch out: log⁡bx\log_b x is never x÷bx \div b.

3. The number e and exponential functions with base e

Warm-up

Three quick questions from earlier lessons.

Discover

Compute (1+1x)x\left(1 + \frac{1}{x}\right)^x for x=1x = 1, 1010, 100100 and 10001000 with a calculator.

What do you notice as xx gets larger?

Key idea

The values approach a number called e≈2.718e \approx 2.718. Functions with base ee describe continuous growth.

Continuous growth at rate rr (as a decimal): A(t)=A0 ertA(t) = A_0 \, e^{rt}

In f(t)=a ektf(t) = a \, e^{kt}, increasing tt by 1 multiplies ff by eke^{k}.

Worked example

The area of a mold is a(t)=10eta(t) = 10e^{t} mm² after tt days. By what factor does it grow each day, and what is the area after 2 days?

Daily factor: e≈2.718e \approx 2.718
Here k = 1, so each day multiplies the area by e¹.
a(2)=10e2a(2) = 10e^{2}
Substitute t = 2.
a(2)≈73.9a(2) \approx 73.9 mm²
e² ≈ 7.389.

Your turn, guided

Your turn: 500 dollars grows continuously at 4% per year. Fill in the missing line.

r=0.04r = 0.04
4% as a decimal.
A(t)=500e0.04tA(t) = 500e^{0.04t}
Use A₀ e^(rt).

Practice ★

  1. An account holds 400 dollars and grows continuously at 3% per year. Write the amount A(t)A(t) after tt years.
  2. An account holds 100 dollars and grows continuously at 8% per year. Write the amount A(t)A(t) after tt years.
  3. An account holds 300 dollars and grows continuously at 3% per year. Write the amount A(t)A(t) after tt years.
  4. An account holds 600 dollars and grows continuously at 6% per year. Write the amount A(t)A(t) after tt years.
  5. An account holds 500 dollars and grows continuously at 5% per year. Write the amount A(t)A(t) after tt years.

Watch out!

Practice ★★

  1. The area of mold is f(t)=2e4tf(t) = 2e^{ 4t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  2. The area of mold is f(t)=3e1tf(t) = 3e^{ 1t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  3. The area of mold is f(t)=7e4tf(t) = 7e^{ 4t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  4. The area of mold is f(t)=2e2tf(t) = 2e^{ 2t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  5. The area of mold is f(t)=3e4tf(t) = 3e^{ 4t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.

Exit ticket

Three questions to check you have got it.

  1. The area of mold is f(t)=2e4tf(t) = 2e^{ 4t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  2. The area of mold is f(t)=3e1tf(t) = 3e^{ 1t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.
  3. The area of mold is f(t)=7e4tf(t) = 7e^{ 4t } square millimetres after tt days. By what factor is the area multiplied each day? Give the exact value.

Summary card

The number ee

  • e≈2.718e \approx 2.718, the limit of (1+1x)x\left(1 + \frac{1}{x}\right)^x.
  • Continuous growth at rate rr: A(t)=A0 ertA(t) = A_0 \, e^{rt}.
  • In a ekta \, e^{kt}, the factor per unit of tt is eke^{k}.

Watch out: "continuously at rr" is not "by 1+r1 + r once a year".

4. Logarithmic functions and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

The graph of y=2xy = 2^x passes through (0,1)(0, 1), (1,2)(1, 2) and (3,8)(3, 8). Swap the coordinates of each point.

Which function has a graph through the swapped points?

Key idea

y=log⁡bxy = \log_b x is the inverse of y=bxy = b^x: its graph is the reflection of y=bxy = b^x in the line y=xy = x.

The graph of y=log⁡bxy = \log_b x passes through (1,0)(1, 0) and (b,1)(b, 1), has domain x>0x > 0 and a vertical asymptote x=0x = 0

For y=log⁡b(x−h)y = \log_b(x - h), everything shifts hh units right: the asymptote is x=hx = h and the graph crosses the xx-axis where x−h=1x - h = 1.

Worked example

Where does y=log⁡2(x−3)y = \log_2(x - 3) cross the xx-axis?

log⁡2(x−3)=0\log_2(x - 3) = 0
On the x-axis, y = 0.
x−3=1x - 3 = 1
A logarithm is 0 when its input is 1.
x=4x = 4
Add 3.

Your turn, guided

Your turn: where does y=ln⁡(x−5)y = \ln(x - 5) cross the xx-axis? Fill in the missing line.

x−5=1x - 5 = 1
ln of the input is 0 when the input is 1.
x=6x = 6
Add 5.

Practice ★

  1. Which statements are true about the graph of y=log⁡9xy = \log_{ 9 } x? Tick all that are true.
    1. It passes through (1, 0).
    2. It passes through (9, 1).
    3. It passes through (0, 1).
    4. It has the horizontal asymptote y = 0.
    5. It is defined only for x > 0.
  2. Which statements are true about the graph of y=log⁡2xy = \log_{ 2 } x? Tick all that are true.
    1. It passes through (1, 0).
    2. It passes through (2, 1).
    3. It passes through (0, 1).
    4. It has the horizontal asymptote y = 0.
    5. It is defined only for x > 0.
  1. The graph of y=log⁡bxy = \log_b x passes through the point (9,2)(9, 2). What is bb?
  2. The graph of y=log⁡bxy = \log_b x passes through the point (81,4)(81, 4). What is bb?
  3. The graph of y=log⁡bxy = \log_b x passes through the point (8,3)(8, 3). What is bb?
  4. The graph of y=log⁡bxy = \log_b x passes through the point (625,4)(625, 4). What is bb?
  5. The graph of y=log⁡bxy = \log_b x passes through the point (64,3)(64, 3). What is bb?

Watch out!

Practice ★★

  1. Here is the graph of y=log⁡3(x−5)y = \log_{ 3 }(x - 5). Tap the point where it crosses the xx-axis.
  2. Here is the graph of y=log⁡2(x−4)y = \log_{ 2 }(x - 4). Tap the point where it crosses the xx-axis.
  1. What is the domain of y=ln⁡(x−6)y = \ln(x - 6)? Give it as an interval.
  2. What is the domain of y=ln⁡(x−9)y = \ln(x - 9)? Give it as an interval.
  3. What is the domain of y=ln⁡(x−7)y = \ln(x - 7)? Give it as an interval.
  4. What is the domain of y=ln⁡(x−2)y = \ln(x - 2)? Give it as an interval.
  5. What is the domain of y=ln⁡(x−8)y = \ln(x - 8)? Give it as an interval.

Exit ticket

Three questions to check you have got it.

  1. Where does the graph of y=ln⁡(x−3)y = \ln(x - 3) cross the xx-axis? Give the value of xx.
  2. Where does the graph of y=ln⁡(x−6)y = \ln(x - 6) cross the xx-axis? Give the value of xx.
  3. Where does the graph of y=ln⁡(x−7)y = \ln(x - 7) cross the xx-axis? Give the value of xx.

Summary card

Logarithmic functions

  • y=log⁡bxy = \log_b x is the inverse of y=bxy = b^x.
  • Through (1,0)(1, 0) and (b,1)(b, 1); domain x>0x > 0; asymptote x=0x = 0.
  • y=log⁡b(x−h)y = \log_b(x - h): asymptote x=hx = h; crosses the xx-axis at x=h+1x = h + 1.

5. Properties of logarithms: product, quotient and power rules

Warm-up

Three quick questions from earlier lessons.

Discover

Use the fact that log⁡2x\log_2 x is the exponent of 2. Compare log⁡24+log⁡28\log_2 4 + \log_2 8 with log⁡232\log_2 32.

Why are they equal? Hint: 4⋅8=324 \cdot 8 = 32, and 22⋅23=22+32^2 \cdot 2^3 = 2^{2+3}.

Key idea

Logarithms turn multiplication into addition, because exponents add when powers multiply.

log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N; log⁡bMN=log⁡bM−log⁡bN\log_b \dfrac{M}{N} = \log_b M - \log_b N; log⁡b(Mp)=plog⁡bM\log_b(M^p) = p \log_b M

These hold for M>0M > 0 and N>0N > 0.

Worked example

Evaluate log⁡64+log⁡69\log_6 4 + \log_6 9 without a calculator.

=log⁡6(4⋅9)= \log_6(4 \cdot 9)
Product rule.

Why is the first line true?

  1. The log of a product is the sum of the logs
  2. You can add inside the logarithm
  3. The logs multiply

=log⁡636= \log_6 36
4 · 9 = 36.
=2= 2
6² = 36.

Your turn, guided

Your turn: evaluate 2log⁡36−log⁡342\log_3 6 - \log_3 4. Fill in the missing line.

=log⁡336−log⁡34= \log_3 36 - \log_3 4
Power rule: 2 log₃ 6 = log₃ 36.
=log⁡39= \log_3 9
Quotient rule: 36 ÷ 4 = 9.
=2= 2
3² = 9.

Practice ★

  1. Which expression is equal to log⁡8(2⋅7)\log_{ 8 }(2 \cdot 7)?
    1. log⁡82+log⁡87\log_{ 8 } 2 + \log_{ 8 } 7
    2. log⁡82⋅log⁡87\log_{ 8 } 2 \cdot \log_{ 8 } 7
    3. log⁡8(2+7)\log_{ 8 }(2 + 7)
    4. 7log⁡82 7 \log_{ 8 } 2
  2. Which expression is equal to log⁡7(4⋅9)\log_{ 7 }(4 \cdot 9)?
    1. log⁡74+log⁡79\log_{ 7 } 4 + \log_{ 7 } 9
    2. log⁡74⋅log⁡79\log_{ 7 } 4 \cdot \log_{ 7 } 9
    3. log⁡7(4+9)\log_{ 7 }(4 + 9)
    4. 9log⁡74 9 \log_{ 7 } 4
  3. Which expression is equal to log⁡7(5⋅3)\log_{ 7 }(5 \cdot 3)?
    1. log⁡75+log⁡73\log_{ 7 } 5 + \log_{ 7 } 3
    2. log⁡75⋅log⁡73\log_{ 7 } 5 \cdot \log_{ 7 } 3
    3. log⁡7(5+3)\log_{ 7 }(5 + 3)
    4. 3log⁡75 3 \log_{ 7 } 5
  1. Evaluate log⁡1020+log⁡105\log_{ 10 } 20 + \log_{ 10 } 5 without a calculator.
  2. Evaluate log⁡1040+log⁡1025\log_{ 10 } 40 + \log_{ 10 } 25 without a calculator.
  3. Evaluate log⁡1050+log⁡1020\log_{ 10 } 50 + \log_{ 10 } 20 without a calculator.
  4. Evaluate log⁡1025+log⁡104\log_{ 10 } 25 + \log_{ 10 } 4 without a calculator.
  5. Evaluate log⁡612+log⁡618\log_{ 6 } 12 + \log_{ 6 } 18 without a calculator.

Watch out!

Practice ★★

  1. Evaluate 2log⁡26−log⁡292\log_{ 2 } 6 - \log_{ 2 } 9 without a calculator.
  2. Evaluate 2log⁡36−log⁡342\log_{ 3 } 6 - \log_{ 3 } 4 without a calculator.
  3. Evaluate 2log⁡515−log⁡592\log_{ 5 } 15 - \log_{ 5 } 9 without a calculator.
  4. Evaluate 2log⁡318−log⁡3122\log_{ 3 } 18 - \log_{ 3 } 12 without a calculator.
  5. Evaluate 2log⁡315−log⁡3752\log_{ 3 } 15 - \log_{ 3 } 75 without a calculator.
  1. Write log⁡3x+2log⁡34\log_{ 3 } x + 2\log_{ 3 } 4 as a single logarithm.
  2. Write log⁡6x+2log⁡65\log_{ 6 } x + 2\log_{ 6 } 5 as a single logarithm.
  3. Write log⁡7x+2log⁡74\log_{ 7 } x + 2\log_{ 7 } 4 as a single logarithm.
  1. Expand log⁡8(x4y5)\log_{ 8 }\left(\dfrac{x^{ 4 }}{y^{ 5 }}\right): write it with logarithms of xx and yy only.
  2. Expand log⁡5(x5y3)\log_{ 5 }\left(\dfrac{x^{ 5 }}{y^{ 3 }}\right): write it with logarithms of xx and yy only.
  3. Expand log⁡8(x2y5)\log_{ 8 }\left(\dfrac{x^{ 2 }}{y^{ 5 }}\right): write it with logarithms of xx and yy only.

Exit ticket

Three questions to check you have got it.

  1. Evaluate 2log⁡26−log⁡292\log_{ 2 } 6 - \log_{ 2 } 9 without a calculator.
  2. Evaluate 2log⁡36−log⁡342\log_{ 3 } 6 - \log_{ 3 } 4 without a calculator.
  3. Evaluate 2log⁡515−log⁡592\log_{ 5 } 15 - \log_{ 5 } 9 without a calculator.

Summary card

Properties of logarithms (M,N>0M, N > 0)

  • Product: log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N.
  • Quotient: log⁡bMN=log⁡bM−log⁡bN\log_b \frac{M}{N} = \log_b M - \log_b N.
  • Power: log⁡b(Mp)=plog⁡bM\log_b(M^p) = p \log_b M.

Watch out: there is no rule for log⁡b(M+N)\log_b(M + N).

6. Solve exponential equations in any base with logarithms

Warm-up

Three quick questions from earlier lessons.

Discover

You can solve 2x=82^x = 8 in your head. What about 2x=72^x = 7?

Which two whole numbers is the answer between? How could logarithms give it exactly?

Key idea

To solve a⋅bx=ca \cdot b^x = c: isolate the power, then take ln⁡\ln of both sides and use the power rule.

bx=k  ⟺  x=ln⁡kln⁡bb^x = k \iff x = \dfrac{\ln k}{\ln b} (for k>0k > 0)

Worked example

Solve 3⋅2x=213 \cdot 2^x = 21.

2x=72^x = 7
Divide both sides by 3 to isolate the power.
xln⁡2=ln⁡7x \ln 2 = \ln 7
Take ln of both sides; power rule.
x=ln⁡7ln⁡2≈2.81x = \dfrac{\ln 7}{\ln 2} \approx 2.81
Divide by ln 2.

Put the steps for solving a⋅bx=ca \cdot b^x = c in order.

  1. Use the power rule: xln⁡b=ln⁡cax \ln b = \ln \frac{c}{a}
  2. Divide by ln⁡b\ln b
  3. Take ln⁡\ln of both sides
  4. Divide both sides by aa

Your turn, guided

Your turn: solve 5x=125^x = 12. Fill in the missing line.

xln⁡5=ln⁡12x \ln 5 = \ln 12
Take ln of both sides; power rule.
x=ln⁡12ln⁡5x = \dfrac{\ln 12}{\ln 5}
Divide by ln 5.

Practice ★

  1. Solve 6x=356^x = 35. Give the exact answer using natural logarithms.
  2. Solve 2x=102^x = 10. Give the exact answer using natural logarithms.
  3. Solve 7x=367^x = 36. Give the exact answer using natural logarithms.
  4. Solve 4x=204^x = 20. Give the exact answer using natural logarithms.
  5. Solve 2x=172^x = 17. Give the exact answer using natural logarithms.

Watch out!

Practice ★★

  1. Solve 2⋅3x=482\cdot 3^x = 48. Give the exact answer.
  2. Solve 6⋅4x=906\cdot 4^x = 90. Give the exact answer.
  3. Solve 4⋅4x=304\cdot 4^x = 30. Give the exact answer.
  4. Solve 6⋅5x=506\cdot 5^x = 50. Give the exact answer.
  5. Solve 2⋅4x=722\cdot 4^x = 72. Give the exact answer.
  1. Solve 4⋅2x=654\cdot 2^x = 65 step by step.
  2. Solve 3⋅5x=633\cdot 5^x = 63 step by step.

Exit ticket

Three questions to check you have got it.

  1. Solve 2⋅3x=482\cdot 3^x = 48. Give the exact answer.
  2. Solve 6⋅4x=906\cdot 4^x = 90. Give the exact answer.
  3. Solve 4⋅4x=304\cdot 4^x = 30. Give the exact answer.

Summary card

Solving a⋅bx=ca \cdot b^x = c

  1. Isolate the power: bx=cab^x = \dfrac{c}{a}.
  2. Take ln⁡\ln: xln⁡b=ln⁡cax \ln b = \ln \dfrac{c}{a}.
  3. Divide: x=ln⁡(c/a)ln⁡bx = \dfrac{\ln(c/a)}{\ln b}.

Watch out: never multiply aa into the base.

7. Exponential models: half-life and doubling time

Warm-up

Three quick questions from earlier lessons.

Discover

A doctor gives a patient 80 mg of a medicine. Every 6 hours, half of what is left leaves the body.

How much is left after 6 hours? After 12? After 24? Does the amount ever reach 0?

Key idea

The half-life hh is the time it takes for an amount to halve; the doubling time dd is the time it takes to double.

Half-life hh: A(t)=A0(12)thA(t) = A_0 \left(\tfrac{1}{2}\right)^{\frac{t}{h}}. Doubling time dd: P(t)=P0⋅2tdP(t) = P_0 \cdot 2^{\frac{t}{d}}

The exponent th\frac{t}{h} counts how many half-lives have passed.

Worked example

A sample of 80 mg has a half-life of 5 years. How much is left after 15 years?

155=3\dfrac{15}{5} = 3 half-lives
Count the half-lives.
A=80(12)3A = 80 \left(\tfrac{1}{2}\right)^3
Halve three times.
A=10A = 10 mg
80 ÷ 8 = 10.

Your turn, guided

Your turn: 96 mg, half-life 2 hours. How much is left after 8 hours? Fill in the missing line.

A=96(12)4A = 96 \left(\tfrac{1}{2}\right)^4
8 ÷ 2 = 4 half-lives.
A=6A = 6 mg
96 ÷ 16 = 6.

Practice ★

  1. A sample of 64 mg of a radioactive substance has a half-life of 2 years. How many mg are left after 6 years?
  2. A sample of 64 mg of a radioactive substance has a half-life of 3 years. How many mg are left after 9 years?
  3. A sample of 96 mg of a radioactive substance has a half-life of 5 years. How many mg are left after 20 years?
  4. A sample of 320 mg of a radioactive substance has a half-life of 10 years. How many mg are left after 40 years?
  5. A sample of 320 mg of a radioactive substance has a half-life of 2 years. How many mg are left after 10 years?

Watch out!

Practice ★★

  1. A sample of 200 mg has a half-life of 2 years. How much is left after 8 years? Round to 2 decimal places.
  2. A sample of 100 mg has a half-life of 3 years. How much is left after 11 years? Round to 2 decimal places.
  3. A sample of 400 mg has a half-life of 3 years. How much is left after 4 years? Round to 2 decimal places.
  4. A sample of 200 mg has a half-life of 3 years. How much is left after 10 years? Round to 2 decimal places.
  5. A sample of 100 mg has a half-life of 4 years. How much is left after 11 years? Round to 2 decimal places.

Exit ticket

Three questions to check you have got it.

  1. A dose of 200 mg of a medicine has a half-life of 2 hours. Write the amount A(t)A(t), in mg, left after tt hours.
  2. A dose of 50 mg of a medicine has a half-life of 3 hours. Write the amount A(t)A(t), in mg, left after tt hours.
  3. A dose of 500 mg of a medicine has a half-life of 4 hours. Write the amount A(t)A(t), in mg, left after tt hours.

Summary card

Half-life and doubling time

  • Half-life hh: A0(12)thA_0 \left(\tfrac{1}{2}\right)^{\frac{t}{h}}. Doubling time dd: P0⋅2tdP_0 \cdot 2^{\frac{t}{d}}.
  • th\frac{t}{h} = number of half-lives.

Watch out: nn half-lives divide by 2n2^n, not by 2n2n.

Answers: Exponential and logarithmic models

1. Interpret exponential functions and their graphs

A

  1. 66
  2. 66

B

  1. Decay: each step multiplies by 14\tfrac{1}{ 4 }

C

  1. 8⋅2x8 \cdot 2^{x}
  2. 7⋅3x7 \cdot 3^{x}

D

  1. 66
  2. 66
  3. 99

2. What is a logarithm? Logarithms as exponents

A

  1. 22
  2. 22
  3. 44
  4. 33
  5. 33

B

  1. log⁡8y=2\log_{ 8 } y = 2 → y=82y = 8^{2} · log⁡8y=3\log_{ 8 } y = 3 → y=83y = 8^{3} · log⁡8y=−1\log_{ 8 } y = -1 → y=18y = \dfrac{1}{ 8 } · log⁡8y=12\log_{ 8 } y = \tfrac{1}{2} → y=8y = \sqrt{ 8 }
  2. log⁡3y=2\log_{ 3 } y = 2 → y=32y = 3^{2} · log⁡3y=3\log_{ 3 } y = 3 → y=33y = 3^{3} · log⁡3y=−1\log_{ 3 } y = -1 → y=13y = \dfrac{1}{ 3 } · log⁡3y=12\log_{ 3 } y = \tfrac{1}{2} → y=3y = \sqrt{ 3 }

C

  1. ln⁡(21)ln⁡(6)\frac{\ln{\left(21 \right)}}{\ln{\left(6 \right)}}
  2. ln⁡(16)ln⁡(7)\frac{\ln{\left(16 \right)}}{\ln{\left(7 \right)}}
  3. ln⁡(31)ln⁡(2)\frac{\ln{\left(31 \right)}}{\ln{\left(2 \right)}}
  4. ln⁡(15)ln⁡(7)\frac{\ln{\left(15 \right)}}{\ln{\left(7 \right)}}
  5. ln⁡(37)ln⁡(2)\frac{\ln{\left(37 \right)}}{\ln{\left(2 \right)}}

D

  1. ln⁡(21)ln⁡(6)\frac{\ln{\left(21 \right)}}{\ln{\left(6 \right)}}
  2. ln⁡(16)ln⁡(7)\frac{\ln{\left(16 \right)}}{\ln{\left(7 \right)}}
  3. ln⁡(31)ln⁡(2)\frac{\ln{\left(31 \right)}}{\ln{\left(2 \right)}}

3. The number e and exponential functions with base e

A

  1. 400e3t100400 e^{\frac{3 t}{100}}
  2. 100e2t25100 e^{\frac{2 t}{25}}
  3. 300e3t100300 e^{\frac{3 t}{100}}
  4. 600e3t50600 e^{\frac{3 t}{50}}
  5. 500et20500 e^{\frac{t}{20}}

B

  1. e4e^{4}
  2. ee
  3. e4e^{4}
  4. e2e^{2}
  5. e4e^{4}

C

  1. e4e^{4}
  2. ee
  3. e4e^{4}

4. Logarithmic functions and their graphs

A

  1. It passes through (1, 0). · It passes through (9, 1). · It is defined only for x > 0.
  2. It passes through (1, 0). · It passes through (2, 1). · It is defined only for x > 0.

B

  1. 33
  2. 33
  3. 22
  4. 55
  5. 44

C

  1. (6,0)(6, 0)
  2. (5,0)(5, 0)

D

  1. (6,∞)(6, \infty)
  2. (9,∞)(9, \infty)
  3. (7,∞)(7, \infty)
  4. (2,∞)(2, \infty)
  5. (8,∞)(8, \infty)

E

  1. 44
  2. 77
  3. 88

5. Properties of logarithms: product, quotient and power rules

A

  1. log⁡82+log⁡87\log_{ 8 } 2 + \log_{ 8 } 7
  2. log⁡74+log⁡79\log_{ 7 } 4 + \log_{ 7 } 9
  3. log⁡75+log⁡73\log_{ 7 } 5 + \log_{ 7 } 3

B

  1. 22
  2. 33
  3. 33
  4. 22
  5. 33

C

  1. 22
  2. 22
  3. 22
  4. 33
  5. 11

D

  1. ln⁡(16x)ln⁡(3)\frac{\ln{\left(16 x \right)}}{\ln{\left(3 \right)}}
  2. ln⁡(25x)ln⁡(6)\frac{\ln{\left(25 x \right)}}{\ln{\left(6 \right)}}
  3. ln⁡(16x)ln⁡(7)\frac{\ln{\left(16 x \right)}}{\ln{\left(7 \right)}}

E

  1. 4ln⁡(x)ln⁡(8)−5ln⁡(y)ln⁡(8)\frac{4 \ln{\left(x \right)}}{\ln{\left(8 \right)}} - \frac{5 \ln{\left(y \right)}}{\ln{\left(8 \right)}}
  2. 5ln⁡(x)ln⁡(5)−3ln⁡(y)ln⁡(5)\frac{5 \ln{\left(x \right)}}{\ln{\left(5 \right)}} - \frac{3 \ln{\left(y \right)}}{\ln{\left(5 \right)}}
  3. 2ln⁡(x)ln⁡(8)−5ln⁡(y)ln⁡(8)\frac{2 \ln{\left(x \right)}}{\ln{\left(8 \right)}} - \frac{5 \ln{\left(y \right)}}{\ln{\left(8 \right)}}

F

  1. 22
  2. 22
  3. 22

6. Solve exponential equations in any base with logarithms

A

  1. ln⁡(35)ln⁡(6)\frac{\ln{\left(35 \right)}}{\ln{\left(6 \right)}}
  2. ln⁡(10)ln⁡(2)\frac{\ln{\left(10 \right)}}{\ln{\left(2 \right)}}
  3. ln⁡(36)ln⁡(7)\frac{\ln{\left(36 \right)}}{\ln{\left(7 \right)}}
  4. ln⁡(20)ln⁡(4)\frac{\ln{\left(20 \right)}}{\ln{\left(4 \right)}}
  5. ln⁡(17)ln⁡(2)\frac{\ln{\left(17 \right)}}{\ln{\left(2 \right)}}

B

  1. ln⁡(24)ln⁡(3)\frac{\ln{\left(24 \right)}}{\ln{\left(3 \right)}}
  2. ln⁡(15)ln⁡(4)\frac{\ln{\left(15 \right)}}{\ln{\left(4 \right)}}
  3. ln⁡(152)ln⁡(4)\frac{\ln{\left(\frac{15}{2} \right)}}{\ln{\left(4 \right)}}
  4. ln⁡(253)ln⁡(5)\frac{\ln{\left(\frac{25}{3} \right)}}{\ln{\left(5 \right)}}
  5. ln⁡(36)ln⁡(4)\frac{\ln{\left(36 \right)}}{\ln{\left(4 \right)}}

C

  1. ln⁡(654)ln⁡(2)\frac{\ln{\left(\frac{65}{4} \right)}}{\ln{\left(2 \right)}}
  2. ln⁡(21)ln⁡(5)\frac{\ln{\left(21 \right)}}{\ln{\left(5 \right)}}

D

  1. ln⁡(24)ln⁡(3)\frac{\ln{\left(24 \right)}}{\ln{\left(3 \right)}}
  2. ln⁡(15)ln⁡(4)\frac{\ln{\left(15 \right)}}{\ln{\left(4 \right)}}
  3. ln⁡(152)ln⁡(4)\frac{\ln{\left(\frac{15}{2} \right)}}{\ln{\left(4 \right)}}

7. Exponential models: half-life and doubling time

A

  1. 88
  2. 88
  3. 66
  4. 2020
  5. 1010

B

  1. 252\frac{25}{2} mg
  2. 25234\frac{25 \sqrt[3]{2}}{4} mg
  3. 100⋅223100 \cdot 2^{\frac{2}{3}} mg
  4. 25⋅2232\frac{25 \cdot 2^{\frac{2}{3}}}{2} mg
  5. 25242\frac{25 \sqrt[4]{2}}{2} mg

C

  1. 200⋅2−t2200 \cdot 2^{- \frac{t}{2}}
  2. 50⋅2−t350 \cdot 2^{- \frac{t}{3}}
  3. 500⋅2−t4500 \cdot 2^{- \frac{t}{4}}