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# Logarithms — Definition, Rules, Examples

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

TL;DR

A logarithm logₓ(b) is the exponent to which the base b must be raised to produce x. This article covers the definition as the inverse of exponentiation, the seven log rules (product, quotient, power, change-of-base, identity, zero, base-of-base), three worked examples, and the historical breakthrough Napier made in 1614.

## The Question "What Power Gives This Number?"

Multiplication makes numbers bigger fast. Exponentiation makes them bigger faster. Logarithms are the inverse — they tell you what power produced a number, taking a huge value and returning the small exponent that built it.

Before pocket calculators, logarithms let astronomers replace huge multiplications with additions of small numbers (a 24-hour calculation turned into a 30-minute one). Today they show up in earthquake magnitudes, pH levels, signal-to-noise ratios, and the algorithms behind data compression. The inverse-of-exponential framing is what makes them everywhere.

## What a Logarithm Is

The **logarithm** of x to base b is the exponent that turns b into x:

logₙ(x)=y; ⟺ by=x.

Three conditions must hold: b>0, b≠1, and x>0.

The most common bases are:

- **Common logarithm** — log(x) usually means log₁₀(x).
- **Natural logarithm** — ln(x) means logₑ(x), where e≈2.71828.
- **Binary logarithm** — log₂(x), used in computer science.

> **Quick facts.**
> - **Definition:** logₙ(x)=y⟺by=x.
> - **Domain:** x>0 (the input to log must be positive).
> - **Range:** all real numbers.
> - **Special values:** logₙ(1)=0, logₙ(b)=1.
> - **Cannot take log of:** zero or negative numbers (over the real numbers).

## The Seven Log Rules of Logarithms

### **1. Product rule**

logₙ(MN)=logₙ(M)+logₙ(N).

The log of a product is the sum of the logs.

### **2. Quotient rule**

logₙ(M/N)=logₙ(M)−logₙ(N).

The log of a quotient is the difference of the logs.

### **3. Power rule**

logₙ(Mᵖ)=p⋅logₙ(M).

The log of a power is the exponent times the log.

### **4. Change-of-base rule**

logₙ(x)=logₗ(x)/logₗ(b) for any base l.

Useful when your calculator only does log₁₀ or ln.

### **5. Identity rule**

logₙ(b)=1 and logₙ(1)=0.

The log of the base is 1; the log of 1 is 0.

### **6. Inverse rules**

b logₙ(x)=x and logₙ(bᵡ)=x.

Exponentiation and logarithm to the same base cancel.

### **7. Equal-arguments rule**

If logₙ(M)=logₙ(N), then M=N.

This is how log equations are solved.

## Three Worked Examples of Logarithms

**Quick.** Evaluate log₂(8).

Ask: what power of 2 gives 8? 2³=8, so log₂(8)=3.

**Final answer:** log₂(8)=3.

**Standard.** Solve log₃(x)+log₃(x−2)=1.

Use the product rule:

log₃(x)+log₃(x−2)=log₃(x(x−2))=1.

Convert to exponential form: x(x−2)=3. Expand to x²−2x−3=0 and factor to get (x−3)(x+1)=0. So x=3 or x=−1. Check the domain: log₃(x) requires x>0, and log₃(x−2) requires x>2, so reject x=−1.

**Final answer:** x=3.

**Stretch.** Solve 5^(x+1)=2⋅3ˣ.

Take ln of both sides:

ln(5^(x+1))=ln(2⋅3ˣ).

Apply log rules:

(x+1)ln(5)=ln(2)+xln(3).

x(ln(5)−ln(3))=ln(2)−ln(5).

x=ln(2/5)/ln(5/3).

Numerically: x≈−1.79.

**Final answer:** x=ln(2/5)/ln(5/3)≈−1.79.

## Where Logarithms Show Up — From Earthquakes to Algorithms

Logarithms compress huge ranges into small ones. That single capability is why they appear in every quantitative field.

- **Richter scale.** Earthquake magnitude is log₁₀ of the seismic amplitude.
- **Decibels.** Sound intensity in decibels is 10 log₁₀(I/I₀).
- **pH scale.** pH=-log₁₀[H⁺].
- **Computer science.** Binary search runs in O(log₂n) time.
- **Information theory.** Shannon entropy uses log₂.
- **Compound interest.** "How long to double money at 5%?" is log(2)/log(1.05)≈14.2 years.

## The Mistakes Students Make Most Often

### **1. Forgetting the domain**

**Where it slips in:** Solving log₃(x)+log₃(x−2)=1 and accepting x=−1.

**Don't do this:** Skip the domain check.

### **2. Trying to take the log of a sum**

**Where it slips in:** Writing log(a+b)=log(a)+log(b).

**Don't do this:** Distribute the log over addition.

### **3. Misapplying the power rule**

**Where it slips in:** log(x²)=(log(x))².

**Don't do this:** Move the exponent inside as a power on the log.

### **4. Trying to take the log of zero or a negative**

**Where it slips in:** Computing log(0) or log(−5).

**Don't do this:** Treat these as defined. **The correct way:** log(0) is undefined. log of a negative number is undefined over the reals.

## The Mathematicians Who Invented Logarithms

**John Napier (1550–1617, Scotland)** published _Mirifici Logarithmorum Canonis Descriptio_ (1614), inventing logarithms.

**Henry Briggs (1561–1630, England)** worked with Napier to develop the base-10 logarithms.

**Leonhard Euler (1707–1783, Switzerland)** introduced the natural logarithm and the constant e in _Introductio in analysin infinitorum_ (1748).

## Conclusion

- A **logarithm** logₙ(x)=y is the exponent that satisfies by=x.
- The seven log rules — product, quotient, power, change-of-base, identity, inverse, equal-arguments — are the algebra of logs.
- The domain is x>0; logs of zero or negative numbers are undefined. 
- The single most common mistake is forgetting to check that candidate solutions to log equations satisfy the original domain.
- Logarithms compress huge ranges into useful scales.