# 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](/content/authors/bhanzu-team/index.html) 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.