Logarithmic Functions

A logarithm answers an exponent question and forms the inverse of an exponential function.

A logarithm reverses exponential growth. A slowly increasing curve lies to the right of a dashed vertical asymptote. The diagonal line y equals x is included as the reflection line used when comparing a logarithm with its exponential inverse.

The statement $\log_b x=y$ means exactly that $b^y=x$.

Definition

For $b>0$, $b\ne1$, and $x>0$,

\log_b x=y\quad\Longleftrightarrow\quad b^y=x.

A logarithm is an exponent. For example,

\log_2 8=3

because $2^3=8$. Translating between logarithmic and exponential form is the central skill.

Inverse relationship

The functions

f(x)=b^x\qquad\text{and}\qquad f^{-1}(x)=\log_b x

undo one another. Their graphs reflect across $y=x$, so exponential domain and range swap for the logarithm.

This gives the inverse identities

\log_b(b^x)=x

and

b^{\log_b x}=x\quad(x>0).

Worked example: evaluate exactly

A negative logarithm output is allowed; only the logarithm's input must be positive.

Transformations

For

y=a\log_b(k(x-d))+c,

the inside restriction $k(x-d)>0$ determines the domain. The parent asymptote $x=0$ transforms to the boundary where $k(x-d)=0$.

Do not state the domain from a memorized shift alone when $k$ is negative; solve the inequality for the logarithm's argument.

Common mistakes

Treating a logarithm as division. It is an exponent.

Allowing zero or negative inputs. Real logarithms require positive arguments.

Splitting a logarithm of a sum. $\log(M+N)$ is not $\log M+\log N$.

Confusing the base and argument. In $\log_bx$, $b$ is the base and $x$ is the positive input.

Forgetting that exponential and logarithmic graphs reflect across $y=x$. Their domain, range, and asymptotes correspond by inversion.

Quick self-check

Can I rewrite the statement in exponential form?
Is the base positive and not equal to $1$?
Is every logarithm argument positive?
Do graph features match the inverse exponential?
Am I using log laws only for products, quotients, and powers?