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Logarithms — Definition, Rules, Examples

#Algebra

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.