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Pure · Exponentials & logarithms

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Logarithms

A logarithm is the inverse of an exponential — it answers "what power?". Converting between log and index form, evaluating logs, the special values, and natural logs (ln).

Pure · Exponentials & logarithms

Logarithms

A logarithm is the inverse of an exponential — it answers "what power?". Converting between log and index form, evaluating logs, the special values, and natural logs (ln).

Why it works

A logarithm answers one question: "what power do I raise the base to?" That is all log⁡\log means — it is the inverse of raising to a power.

log⁡ax=y⟺ay=x(a>0, a≠1, x>0).\log_a x = y \quad \Longleftrightarrow \quad a^y = x \qquad (a > 0,\ a \neq 1,\ x > 0).

Read the left side as "the power that turns aa into xx". For example log⁡28=3\log_2 8 = 3 because 23=82^3 = 8. Being able to flip between the two forms is the whole skill — most log problems are solved by rewriting in index form, or the reverse.

Because they undo each other:

alog⁡ax=xandlog⁡a(ax)=x.a^{\log_a x} = x \qquad\text{and}\qquad \log_a (a^x) = x.

Two special values fall straight out of the definition, and you should know them cold:
  • log⁡a1=0\log_a 1 = 0 (since a0=1a^0 = 1),
  • log⁡aa=1\log_a a = 1 (since a1=aa^1 = a).
Also note you can only take the log of a positive number — there is no power of a positive base that gives 00 or a negative result. That domain restriction is what forces you to reject some "solutions" later on.

Natural logarithms. The log to base ee is written ln⁡\ln: ln⁡x\ln x means log⁡ex\log_e x. It is the inverse of exe^x, so

ln⁡(ex)=x,eln⁡x=x,ln⁡e=1,ln⁡1=0.\ln(e^x) = x, \qquad e^{\ln x} = x, \qquad \ln e = 1, \qquad \ln 1 = 0.

A "log⁡\log" with no base written usually means base 1010 (log⁡10\log_{10}), the button on your calculator next to ln⁡\ln.