Logarithm Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Logarithm.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The logarithm \log_b(x) answers: "to what power must b be raised to produce x?" It is the inverse function of f(x) = b^x.

The exponent that produces a number. \log_2(8) = 3 because 2^3 = 8.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Logarithms turn multiplication into addition: \log(ab) = \log(a) + \log(b).

Common stuck point: \log without a base usually means \log_{10} (common) or \log_e (natural).

Sense of Study hint: Rewrite the log as a question: log base b of x means 'b to what power equals x?' Then guess and check.

Worked Examples

Example 1

easy
Evaluate \log_2 32.

Solution

  1. 1
    A logarithm asks for the exponent, so we want the value of x such that 2^x = 32.
  2. 2
    Check powers of 2: 2^5 = 32.
  3. 3
    Therefore \log_2 32 = 5.

Answer

5
A logarithm answers the question: 'What power do I raise the base to in order to get this number?' The definition \log_b a = c means b^c = a.

Example 2

medium
Solve \log_3(2x + 1) = 4.

Example 3

hard
Solve \log_2(x) + \log_2(x - 6) = 4.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Evaluate \log_5 125.

Example 2

medium
Solve \log(x) + \log(x - 3) = 1 where \log denotes \log_{10}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

exponential functioninverse function