Logarithms Worksheets

Evaluate logarithms and solve logarithmic equations by rewriting them in exponential form. Grades 11–12.

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What is a logarithm?

A logarithm answers the question "what exponent do I need?". The statement log⁡b(y)=n\log_{b}(y) = n means exactly the same as bn=yb^{n} = y. For example, log⁡2(8)=3\log_{2}(8) = 3 because 23=82^{3} = 8. When no base is written, as in log⁡(1000)\log(1000), the base is 10.

Every problem on these worksheets is solved with that one definition: switch between logarithmic form and exponential form, whichever makes the problem easier.

  • Easy — evaluate log⁡b(y)\log_{b}(y) where yy is a power of the base (bases 2, 3, 5 and 10).
  • Medium — solve log⁡b(x)=n\log_{b}(x) = n for xx.
  • Hard — solve log⁡b(ax+c)=n\log_{b}(ax + c) = n, which ends with a linear equation.

Worked examples

Easy: log⁡3(81)\log_{3}(81)

Ask: 3 to what power is 81? Since 34=813^{4} = 81, the answer is log⁡3(81)=4\log_{3}(81) = 4.

Medium: log⁡5(x)=3\log_{5}(x) = 3

Rewrite in exponential form: the base 5 raised to the value 3 gives xx.

x=53=125x = 5^{3} = 125

Hard: log⁡2(3x+2)=3\log_{2}(3x + 2) = 3

Exponential form gives a linear equation:

3x+2=23=8⇒3x=6⇒x=23x + 2 = 2^{3} = 8 \quad\Rightarrow\quad 3x = 6 \quad\Rightarrow\quad x = 2

Check your answers

Substitute back: with x=2x = 2, 3x+2=83x + 2 = 8 and log⁡2(8)=3\log_{2}(8) = 3. The expression inside a logarithm must be positive, and here it is, so x=2x = 2 is a valid solution.

Common mistakes

  • Dividing instead of finding the exponent. log⁡3(81)\log_{3}(81) is not 81÷3=2781 \div 3 = 27.
  • Swapping base and exponent. log⁡5(x)=3\log_{5}(x) = 3 gives x=53=125x = 5^{3} = 125, not 35=2433^{5} = 243. The base of the logarithm stays the base of the power.
  • Setting the inside equal to the right side. In the hard example, 3x+2=33x + 2 = 3 is wrong; the inside equals 232^{3}.
  • Forgetting the hidden base. log⁡(x)=3\log(x) = 3 means base 10, so x=1000x = 1000.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

  1. log⁡2(4)=\log_{2}(4) =
  2. log⁡3(9)=\log_{3}(9) =
  3. log⁡(10)=\log(10) =

Medium

  1. log⁡3(x)=4\log_{3}(x) = 4
    Solve for x
  2. log⁡3(x)=3\log_{3}(x) = 3
    Solve for x
  3. log⁡2(x)=3\log_{2}(x) = 3
    Solve for x

Hard

  1. log⁡5(5x)=2\log_{5}(5x) = 2
    Solve for x
  2. log⁡2(2x+2)=2\log_{2}(2x + 2) = 2
    Solve for x
  3. log⁡3(4x+5)=2\log_{3}(4x + 5) = 2
    Solve for x
Show answers
  1. 2
  2. 2
  3. 1
  4. x = 81
  5. x = 27
  6. x = 8
  7. x = 5
  8. x = 1
  9. x = 1

Tips for teachers and parents

  • Students who find logarithms hard usually need the powers of 2, 3, 5 and 10 at their fingertips. A quick powers table before the worksheet helps.
  • Ask students to write the exponential form of every problem on its own line, even on Easy. It is the whole method.
  • Pair this topic with Exponential Equations: logarithms are the tool for exponential equations whose sides cannot be written with the same base.

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