Logarithms as Inverses

2ˣ = 8 asks what power of 2 gives 8. The answer, 3, is written log₂ 8.

An inverse

A logarithm undoes an exponential, exactly as division undoes multiplication.

The laws

log(ab) = log a + log b, log(a/b) = log a − log b, and log(aⁿ) = n log a.

Where the laws come from

Multiplying powers adds exponents. A logarithm *is* an exponent, so it inherits that rule.

A logarithm asks "what power?"

log₂ 8 = 3 because 2³ = 8. The logarithm is the exponent you need. Reading every log statement back as an exponential statement is the single most useful habit in this chapter.

Logs and exponentials are inverses

The log function undoes the exponential with the same base, so their graphs are reflections in the line y = x. Everything true of inverse functions applies, including the swapping of domain and range.

You cannot take the log of zero or a negative

No power of a positive base gives a negative number or zero, so the domain of a log is the positive reals only. Solutions to log equations must be checked against this, and some will be discarded.

The two bases you will meet most

Base 10 is written log and base e is written ln. Base e appears wherever growth is continuous, which is most of science and finance. The change-of-base formula converts between any two bases when needed.