What is a logarithm? Logarithms as exponents

About this lesson

Precalculus › Exponential and logarithmic functions

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

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.

End of the lesson

Next lesson: Asymptotes and shifts of exponential and logarithmic graphs Back to the chapter

Answers: 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)}}

CC BY-SA 4.0