Logarithms and inverse structure
Objective. Translate between exponential and logarithmic statements.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A logarithm answers the question 'to what power must the base be raised to give this number?', making it the inverse operation of exponentiation. If bˣ = y then logb(y) = x, so logarithmic and exponential statements describe the same relationship from opposite directions. For example, since 2⁴ = 16, we know log2(16) = 4. The base must be positive and not equal to 1, and logarithms are only defined for positive arguments. Switching fluently between the two forms, logb(y) = x ⇔ bˣ = y, is the key skill for solving equations involving unknown powers.
Worked example
Write the statement 3⁵ = 243 using logarithms, then use this to evaluate log3(243).
- 1Identify base 3, power 5, result 243.
- 2Rewrite as a logarithm: log3(243) = 5.
- 3Check by computing 3⁵ = 3×3×3×3×3 = 243.
- 4Since this matches, log3(243) = 5 is confirmed.
Answer: log3(243) = 5
Concept mastery
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Practice
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Start practisingCommon mistake. A common error is writing log2(16) = 8 by dividing 16 by 2 instead of asking what power of 2 gives 16. Always rewrite as an exponential equation first: 2ˣ = 16 gives x = 4, not a division.
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