Logarithmic Functions

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

Definition

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

A logarithm is an exponent. For example,

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

Inverse relationship

The functions

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

This gives the inverse identities

and

Graph features

For $y=\log_b x$:

• domain: $x>0$;
• range: all real numbers;
• $x$-intercept: $(1,0)$ because $\log_b1=0$;
• vertical asymptote: $x=0$;
• no $y$-intercept.

If $b>1$, the function increases. If $0<b<1$, it decreases.

Key values

Powers of the base become anchor points. For base $3$:

Thus the graph contains $(1,0)$, $(3,1)$, $(9,2)$, and $(1/3,-1)$.

Worked example: evaluate exactly

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

Logarithm laws

For positive $M,N$:

These follow from exponent laws. There is no corresponding law that splits $\log_b(M+N)$.

Common and natural logarithms

$\log x$ usually means base $10$, while $\ln x$ means base $e$, where $e\approx2.71828$. Calculators often provide dedicated keys for both.

Any positive base other than $1$ can be evaluated using change of base:

Transformations

For

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.

Logarithmic scales

Logarithms compress wide ranges. Decibels, pH, earthquake magnitude, and information measures use logarithmic ideas. A one-unit change can represent multiplication by a fixed factor rather than addition of a fixed amount.

Interpret the base and scale definition carefully; different logarithmic scales use different multipliers and reference values.

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?

Related topics

• Exponential Functions
• Logarithmic Equations
• Inverse Functions
• Exponents