The big idea: A logarithm answers the question "what power?".
log_a b is the power you raise a to, to get b.
For example, log₂ 8 = 3, because 2³ = 8.
- the base (a > 0, a ≠ 1)
- the result (b > 0)
- the power — i.e. the logarithm
Logs undo powers: ax = b and x = loga b are the same fact written two ways.
Taking a log undoes raising to a power, and vice versa — they are inverses.
Same fact, two forms: To convert, keep the base as the base; the log equals the exponent:
ax = b ⇔ loga b = x.
IB-style question — exponent → log
Write 3⁴ = 81 in logarithmic form.
Step by step
- The base 3 stays the base; the exponent 4 becomes the value of the log.
Final answer
log₃ 81 = 4.
IB-style question — log → exponent
Write log₅ 125 = 3 in exponent form.
Step by step
- The base 5 stays the base; the log value 3 is the exponent.
Final answer
5³ = 125.
Spot the base: The small subscript is the base — and it is the base in both forms. The log value is always the power.
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Two logs written without a base: log x means log₁₀ x (base 10).
ln x means log_e x (base e, where e ≈ 2.718).
They are just logs with a hidden base.
Quick values: log 1000 = 3 (because 10³ = 1000).
ln e = 1 (because e¹ = e), and ln 1 = 0.