Logarithms: the inverse of exponentials

Exponentials raise a base to a power. Logarithms run that backward: given the result, they recover the power. A logarithm is the inverse of an exponential — exactly the idea from Module 1, now made concrete.

"What power?"

The logarithm answers one question:

logᵦ(x) = y means bʸ = x — "to what power must I raise b to get x?"

Read these both ways:

• log₂(8) = 3, because 2³ = 8.
• log₁₀(1000) = 3, because 10³ = 1000.
• log₅(1) = 0, because 5⁰ = 1.

The base b is the same base you'd use in the exponential. Two bases are so common they get shorthand: log x means base 10, and ln x (the natural log) means base e.

The graph: an exponential in the mirror

Because logs invert exponentials, their graphs are reflections across y = x — the very picture from the inverse-functions lesson:

This reflection explains every feature of the log graph at a glance:

• The exponential's point (0, 1) becomes the log's (1, 0): logᵦ(1) = 0 always.
• The exponential's horizontal asymptote y = 0 becomes the log's vertical asymptote x = 0.
• The domain of a log is x > 0 — you can't take the log of zero or a negative number, because no power of a positive base lands there.

They cancel

As inverses, an exponential and a log with the same base undo each other:

• logᵦ(bˣ) = x
• b^(logᵦ x) = x

This is the move that solves equations with x trapped in an exponent — take the log of both sides and the exponent comes down.

Why this matters

Logs turn multiplication into addition (next lesson) and tame numbers that span huge ranges. The decibel scale, pH, the Richter scale, and the "orders of magnitude" scientists reason in are all logarithmic — because a log converts "how many times bigger" into a plain, addable distance.