Logarithmic Functions - Graph, Properties, Examples

TL;DR
A logarithmic function f(x)=log⁡b(x) is the inverse of an exponential function — it answers "what power must I raise b to in order to get x?" The two most common log functions are log⁡(x) (base 10) and ln⁡(x) (base e). The graph passes through (1,0), grows without bound but slowly, has domain (0,∞), and has a vertical asymptote at x=0.

BT
Bhanzu Team Last updated on May 19, 2026 6 min read

What Is a Logarithmic Function?

A logarithmic function is a function of the form:
f(x)=log⁡b(x)
where b is the base (b>0, b≠1). It's defined by the inverse relationship:
y=log⁡b(x)⟺b^y=x
In words: log⁡b(x) is "the exponent you raise b to in order to get x."

The two most-used logarithmic functions:

• Common logarithm: log⁡(x)=log⁡10(x) — base 10. Often written without a subscript.
• Natural logarithm: ln⁡(x)=log⁡e(x) — base e≈2.71828. The default in calculus.

The Graph of a Logarithmic Function

For f(x)=log⁡b(x) with b>1:

• Passes through (1,0) — because log⁡b(1)=0 for any base.
• Passes through (b,1) — because log⁡b(b)=1.
• Domain: (0,∞) — only positive inputs.
• Range: (−∞,∞) — all real outputs.
• Vertical asymptote at x=0 — the curve approaches the y-axis but never touches it. As x→0+, f(x)→−∞.
• Increasing for b>1 — grows monotonically but very slowly.

For 0<b<1, the graph flips vertically — decreasing instead of increasing, but with the same domain/range/asymptote.

Properties of Logarithmic Functions

The behaviour of log⁡b is governed by the same identities that hold for logarithms in general.

Algebraic Properties

• log⁡b(xy)=log⁡b(x)+log⁡b(y) (product rule)
• log⁡b(x/y)=log⁡b(x)−log⁡b(y) (quotient rule)
• log⁡b(x^n)=n log⁡b(x) (power rule)
• log⁡b(1)=0, log⁡b(b)=1

Function-Specific Properties

• f(x⋅y)=f(x)+f(y) (turns multiplication into addition)
• f(x^n)=n⋅f(x) (scales by exponent)
• f−1(x)=b^x (inverse is exponential)

Calculus Properties (for f(x)=ln⁡(x))

• ddx(ln⁡(x))=1/x, x>0
• ∫ln(x)dx=xln(x)−x+C

Three Worked Examples — Quick, Standard, Stretch

Quick — Evaluate a Log

Compute log⁡2(32).

Ask: 2?=32. Since 2^5=32, the answer is log⁡2(32)=5.

Standard — Solve a Log Equation

Solve log⁡3(x)=4 for x.

By definition: log⁡3(x)=4⟺x=3^4=81.

Check. log⁡3(81)=4 ✓.

Stretch — Graph Transformation

Sketch f(x)=log⁡2(x−3)+1.

Key points:

• The graph shifts right by 3 and up by 1.
• Domain: (3,∞). Vertical asymptote: x=3.

Why Do Logarithmic Functions Matter? (The Real-World GROUND)

Logarithmic functions describe how the natural world compresses and expands across enormous scales.

• Richter scale (earthquakes): Magnitude is logarithmic.
• Decibel scale (sound): Loudness in decibels is L=10log⁡10(I/I0).
• pH scale (acidity): pH=−log⁡10[H+].
• Information theory: The information content is −log⁡2(p).
• Population growth: Exponential growth has its natural inverse in logarithmic form.

The logarithmic function was invented in 1614 by John Napier to simplify multiplication of large numbers.

A Worked Example — Wrong Path First

Solve log⁡2(x+1)=3.

The correct method: log⁡2(x+1)=3⟺x + 1=2^3=8, so x=7.

Check. log⁡2(7+1)=log⁡2(8)=3 ✓.

What Are the Most Common Mistakes With Logarithmic Functions?

Mistake 1: Taking the log of a non-positive number

Correct way: log⁡b(x) is undefined for x≤0.

Mistake 2: Confusing log⁡(x^2) with (log⁡x)^2

Correct way: log⁡2(x^2)=2log⁡2(x).

Mistake 3: Forgetting the asymptote

Correct way: The y-axis is a vertical asymptote.

Key Takeaways

• A logarithmic function f(x)=log⁡b(x) is the inverse of b^x.
• Domain: (0,∞); range: (−∞,∞); passes through (1,0).
• Vertical asymptote at x=0 — the curve never crosses the y-axis.

A Practical Next Step

Try these three before moving on to exponential growth and decay.

1. Compute log⁡5(125).
2. Solve log⁡2(x)=5 for x.
3. Sketch the graph of f(x)=ln(x+2).

If problem 3 felt tricky, ln(x+2) shifts the natural log graph left by 2 — vertical asymptote at x=−2.