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Logarithmic Functions

Understand logarithmic graphs, domain restrictions, and change of base. Three worked examples cover comparisons, simplification, and a variable base.

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Tutorial Content

Logarithmic Functions

A logarithm answers a question about an exponent. For example, $\log_2 8=3$ because $2^3=8$. This connection lets you move between logarithmic and exponential equations:

For real logarithms, the base must satisfy $a>0$ and $a\ne1$, and the argument must be positive. In this tutorial, $\lg x$ means $\log_{10}x$, while $\ln x$ means $\log_e x$.

1. Reading the graph

The parent function $y=\log_a x$ has domain $(0,+\infty)$ and range $\mathbb{R}$. It passes through $(1,0)$ because $a^0=1$, and through $(a,1)$ because $a^1=a$. The line $x=0$ is a vertical asymptote; the graph never meets the $y$-axis.

Its direction depends on the base:

• If $a>1$, the function is increasing.
• If $0<a<1$, the function is decreasing.

The following exact values show the difference without relying on a sketch.

For $a>1$, $\log_a x$ tends to $-\infty$ as $x\to0^+$ and to $+\infty$ as $x\to+\infty$. For $0<a<1$, these limits are reversed.

The functions $y=\log_a x$ and $y=a^x$ are inverses. Their graphs are reflections in the line $y=x$: a point $(u,v)$ on one corresponds to $(v,u)$ on the other.

These properties describe the parent function. For an expression such as $\log_a(x-2)$, check the whole argument: the domain is $x>2$, and the vertical asymptote is $x=2$.

2. Using logarithm laws

For $M>0$, $N>0$, $a>0$, $a\ne1$, and a real number $r$,

The positivity conditions matter. For example, $\log_a(x^2)$ is defined when $x\ne0$, but $2\log_a x$ is defined only when $x>0$. An identity valid for every nonzero real $x$ is

There is no general rule that splits a logarithm of a sum. Taking $M=N=1$ gives $\log_a(M+N)=\log_a2$, whereas $\log_a M+\log_a N=0$.

3. Changing the base

To express a logarithm in a convenient base $b$, use

Two useful consequences are

for valid bases $a,b$ and $c>0$, and

for $a>0$, $a\ne1$, $b>0$, real $m$, and real $n\ne0$. The last condition is necessary: $n=0$ would make the base equal to $1$.

4. Worked examples

Example 1: Compare logarithms with the same base

Compare $\log_2 3$ with $\log_2 5$, then compare $\log_{0.5}3$ with $\log_{0.5}5$.

Since $2>1$, $\log_2 x$ is increasing:

Since $0<0.5<1$, $\log_{0.5}x$ is decreasing:

When comparing logarithms with the same valid base, check whether the function increases or decreases before choosing the inequality sign.

Example 2: Simplify a product

Evaluate

Changing the bases to $2$ and $3$ gives

Because $\log_2 3\;\log_3 2=1$, the product is

Example 3: Find the domain when the base also varies

Find the domain of $y=\log_{2x-1}(3-x)$.

Apply all three restrictions:

These give $x>\frac12$, $x\ne1$, and $x<3$. Their intersection is

Checking only the argument would miss both restrictions on the base.

5. Before you finish

Check the base and argument in the original expression, not just in a simplified form. Keep any excluded values when you transform an equation. For inequalities with a fixed base between $0$ and $1$, remember that applying the logarithm reverses the order.

Further reading

• OpenStax: Graphs of Logarithmic Functions
• OpenStax: Logarithmic Properties

Content reviewed: 9 September 2026.