# Properties of Logarithms - Laws, Formulas, Proofs

[Algebra](/content/tag/algebra/index.html)

TL;DR

The properties of logarithms are seven identities that simplify logarithmic expressions: the product rule (\( \log_b(xy) = \log_b x + \log_b y \)), quotient rule (\( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y \)), power rule (\( \log_b(x^n) = n \log_b x \)), change-of-base formula, and three special values (\( \log_b 1 = 0 \), \( \log_b b = 1 \), \( \log_b b^n = n \)).

MV

[Madhukar V](/content/authors/madhukar-v/index.html) Last updated on May 15, 2022 9 min read

## What Is a Logarithm? (Quick Refresher)

A **logarithm** answers the question _"to what power must I raise the base to get this number?"_ If \( b^y = x \), then \( \log_b(x) = y \). Example: \( \log_2(8) = 3 \) because \( 2^3 = 8 \).

The two most common bases are:

- **Common logarithm** — base 10, often written as \( \log(x) \) with no subscript
- **Natural logarithm** — base \( e \approx 2.71828 \), written \( \ln(x) \)

Every property below holds for any base \( b \) (with \( b > 0 \) and \( b \neq 1 \)).

## What Are the Properties of Logarithms?

The seven core properties, in compact form:

| Property | Identity | When You Use It |
| --- | --- | --- |
| **Product** | \( \log_b(xy) = \log_b x + \log_b y \) | Multiplication → addition |
| **Quotient** | \( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y \) | Division → subtraction |
| **Power** | \( \log_b(x^n) = n \log_b x \) | Exponent → coefficient |
| **Change of base** | \( \log_b x = \frac{\log_c x}{\log_c b} \) | Converting between bases |
| **Log of 1** | \( \log_b 1 = 0 \) | Anything to the 0 is 1 |
| **Log of base** | \( \log_b b = 1 \) | Anything to the 1 is itself |
| **Log of base power** | \( \log_b(b^n) = n \) | Inverse of exponential |

## How Do the Three Main Properties Work? (With Proofs)

### Product Rule

\( \log_b(xy) = \log_b x + \log_b y \)

**Proof.** Let \( m = \log_b x \) and \( n = \log_b y \). Then \( x = b^m \) and \( y = b^n \), so:

\[ xy = b^m \cdot b^n = b^{m+n} \]

Taking \( \log_b \) of both sides: \( \log_b(xy) = m + n = \log_b x + \log_b y \). ∎

**Example.** \( \log_2(8 \times 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5 \).

### Quotient Rule

\( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y \)

**Proof.** Same setup as above: \( \frac{x}{y} = \frac{b^m}{b^n} = b^{m-n} \), so \( \log_b\left(\frac{x}{y}\right) = m - n \). ∎

**Example.** \( \log_3\left(\frac{81}{9}\right) = \log_3 81 - \log_3 9 = 4 - 2 = 2 \).

### Power Rule

\( \log_b(x^n) = n \log_b x \)

**Proof.** Let \( m = \log_b x \), so \( x = b^m \). Then \( x^n = (b^m)^n = b^{mn} \), giving \( \log_b(x^n) = mn = n \log_b x \). ∎

**Example.** \( \log_2(8^3) = 3 \log_2 8 = 3 \times 3 = 9 \).

## How Do You Use the Change-of-Base Formula?

Calculators only compute \( \log \) (base 10) and \( \ln \) (base e). For any other base, use the **change-of-base formula:**

\[ \log_b x = \frac{\log_c x}{\log_c b} \]

where \( c \) is any base your calculator supports.

**Example.** Compute \( \log_5(125) \) using base-10 logs.

\[ \log_5(125) = \frac{\log 125}{\log 5} \approx 3 \]

Check: \( 5^3 = 125 \) ✓.

**Special case — change to natural log:**

\[ \log_b x = \frac{\ln x}{\ln b} \]

## What Are the Three Special-Value Properties?

These follow directly from the definition of a logarithm.

### Log of 1 = 0

\( \log_b 1 = 0 \) (any base \( b > 0 \), \( b \neq 1 \)).

Because \( b^0 = 1 \) for any non-zero base \( b \).

### Log of the Base Itself = 1

\( \log_b b = 1 \)

Because \( b^1 = b \).

### Log of a Power of the Base = the Exponent

\( \log_b(b^n) = n \)

This is the inverse-function property.

## What Are the Properties of the Natural Logarithm (ln)?

The **natural logarithm** \( \ln(x) \) is just \( \log_e(x) \). Every property above carries over verbatim with \( e \); we restate them here because \( \ln \) is the form you'll meet most often in calculus, physics, and finance.

| Property | Identity for \( \ln \) |
| --- | --- |
| **Product** | \( \ln(xy) = \ln x + \ln y \) |
| **Quotient** | \( \ln\left(\frac{x}{y}\right) = \ln x - \ln y \) |
| **Power** | \( \ln(x^n) = n \ln x \) |
| **Change of base** | \( \ln x = \frac{\log x}{\log e} \) |
| **ln of 1** | \( \ln 1 = 0 \) |
| **ln of e** | \( \ln e = 1 \) |
| **ln of e^n** | \( \ln(e^n) = n \) |
| **Exponential inverse** | \( e^{\ln x} = x \) for all \( x > 0 \) |

**Calculus identity (bonus).** The natural log is the unique logarithm whose derivative is \( \frac{1}{x} \):

\[ \frac{d}{dx}\ln x = \frac{1}{x}, \quad x > 0 \]

## Why Were Logarithms Invented? (The Real-World GROUND)

> _"Seeing there is nothing… that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications…"_ — John Napier, 1614.

Logarithms exist because of a specific 17th-century problem: astronomers and navigators needed to multiply enormous numbers, and multiplication by hand took hours. John Napier published _Mirifici Logarithmorum Canonis Descriptio_ in 1614. His logarithm tables converted multiplication into addition.

Today, logarithmic scales describe enormous ranges of natural phenomena:

- The Richter scale — magnitude-7 earthquake releases 10× the energy of magnitude-6.
- The **decibel scale** — sound intensity doubles roughly every 3 dB.
- The **pH scale** — pH 3 is 10× more acidic than pH 4.

## A Worked Example

Simplify \( \log_2(8 \cdot 4) - \log_2 4 \).

**The intuitive (wrong) approach.** A student applies the product rule incorrectly:

\( \log_2(8 \cdot 4) = \log_2 8 \cdot \log_2 4 = 3 \cdot 2 = 6 \).

Then \( 6 - \log_2 4 = 6 - 2 = 4 \).

**The correct method.**

\( \log_2(8 \cdot 4) - \log_2 4 = \log_2 8 = 3 \).

## What Are the Most Common Mistakes With Properties of Logarithms?

### Mistake 1: Splitting \( \log(x+y) \) as \( \log x + \log y \)

**Where it slips in:** Trying to "distribute" the log over an addition.

**Don't do this:** \( \log(x+y) = \log x + \log y \). **This is false.**

### Mistake 2: Confusing \( \log(x^n) \) with \( (\log x)^n \)

**Where it slips in:** \( \log(x^2) \) and \( (\log x)^2 \) look similar but mean different things.

**Don't do this:** Treating \( (\log x)^2 \) and \( \log(x^2) \) as equal.

### Mistake 3: Treating \( \log 0 \) as 0

**Where it slips in:** Recalling that \( \log 1 = 0 \).

**Don't do this:** Writing \( \log 0 = 0 \).

## The Mathematicians Who Shaped Logarithms

**John Napier (1550–1617, Scotland)**  — Invented logarithms in 1614.

**Henry Briggs (1561–1630, England)**  — Developed the base-10 logarithm tables.

**Leonhard Euler (1707–1783, Switzerland)**  — Established the connection between logarithms and the number \( e \).

## A Practical Next Step

1. Simplify \( \log 100 + \log 10 \) using the product rule.
2. Simplify \( \log_2(64) - \log_2(8) \) using the quotient rule.
3. Compute \( \log_3(81) \) — without a calculator.

## Frequently Asked Questions

What are the 7 properties of logarithms?  
The seven core properties: product, quotient, power, change of base, log of 1, log of base, and log of base power.

What is the change-of-base formula?  
\( \log_b x = \frac{\log_c x}{\log_c b} \) for any valid base \( c \).

Is \( \log(x+y) = \log x + \log y \)?  
No. This is the most common log mistake.

Why is \( \log_b 1 = 0 \)?  
Because \( b^0 = 1 \) for any non-zero base \( b \).

What is the difference between \( \log \) and \( \ln \)?  
\( \log \) usually means base-10 logarithm, and \( \ln \) means base-e logarithm.

How are the properties of logarithms used in real life?  
Logarithmic scales describe earthquake magnitude, sound intensity, acid concentration, and more.