Logarithms
The reverse question of powers
What is this about?
A logarithm answers the question: to what exponent must I raise the base to get this number? log₂(8) asks "2 to the power of what gives 8?". Since 2³ = 8, the answer is 3. The logarithm isn't a new operation: it's a power seen from the other side.
The rule to remember
log_b(N) = x means exactly b^x = N. Facing any logarithm, translate it to its power: the base to the power of what gives N? If the number is an exact power of the base, the answer is immediate.
Example 1
log₃(81)
- 1Translate the question: 3 to the power of what gives 81?
- 2Try powers of 3: 3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81.
- 3The exponent that works is 4.
- 4So log₃(81) = 4. Check: 3⁴ = 81. ✓
Result: 4
Example 2
log(1000) (base 10)
- 1When no base is written, it's 10: 10 to the power of what gives 1000?
- 210¹ = 10, 10² = 100, 10³ = 1000.
- 3The exponent is 3, so log(1000) = 3.
- 4Trick for base 10: count the zeros of the power of 10.
Result: 3
The most common mistake
Many people write
log₂(8) = 4 because 8 ÷ 2 = 4
Why it's wrong
The logarithm is NOT a division. It asks for the EXPONENT: 2 to the power of what gives 8? Since 2³ = 8, the result is 3. Always check by raising the base to your result: 2⁴ = 16 ≠ 8, but 2³ = 8. ✓
Did it click?
The best way to find out is to try. These exercises are exactly on this topic.
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