Logarithms- Properties, Rules, and Problem Solving

What Is a Logarithm?

A logarithm answers a simple question: what exponent produces a given number?

If 2³ = 8, then log₂(8) = 3. The base is 2, the result is 3.

That's it. That's the whole concept. Everything else in logarithms is just rules built around this idea.

You'll see two common bases:

• log with no subscript means base 10 (common logarithm)
• ln means natural logarithm, base e (≈2.718)
• logₐ(x) means base a, whatever number you specify

The Three Core Logarithm Rules

These are the only rules you need. Memorize them. They're not suggestions—they're mathematical law.

1. Product Rule

log_a(xy) = log_a(x) + log_a(y)

When multiplying inside a log, you split it into addition.

2. Quotient Rule

log_a(x/y) = log_a(x) − log_a(y)

When dividing inside a log, you split it into subtraction.

3. Power Rule

log_a(xⁿ) = n · log_a(x)

When raising inside a log to a power, the power comes down as a multiplier.

Change of Base Formula

Sometimes you need to calculate a log with a calculator that only has log₁₀ or ln. Use this:

log_a(x) = log₁₀(x) / log₁₀(a)

Or equivalently with natural log:

log_a(x) = ln(x) / ln(a)

This works for any base. It's your bridge between what you need and what your calculator gives you.

Logarithm Properties at a Glance

Solving Logarithmic Equations

When you see log_a(x) = y, rewrite it as a^y = x. That's the core move.

Example: log₂(x) = 5

Convert: 2⁵ = x

x = 32

Example: log₃(x + 1) = 4

Convert: 3⁴ = x + 1

81 = x + 1

x = 80

Solving Exponential Equations with Logs

When the variable is in the exponent, take the log of both sides.

Example: 5^x = 27

Take log₁₀ of both sides:

log(5^x) = log(27)

Apply power rule:

x · log(5) = log(27)

x = log(27) / log(5)

x ≈ 2.047

You can use any base log—ln, log₁₀, it doesn't matter. The ratio stays the same.

Common Mistakes to Avoid

• log(x + y) ≠ log(x) + log(y). Only works for multiplication, not addition.
• log(x)/log(y) ≠ log_x(y). The correct change of base is log(x)/log(y) = log_y(x). Watch your order.
• Domain restrictions: you cannot take log of zero or a negative number. Ever.
• Forgetting parentheses: log(x²) ≠ 2·log(x) if x could be negative. The power rule still applies, but the original expression may not be defined for negative x.

Getting Started: Practice Problems

Work through these. Don't just read them—write them out.

1. Simplify: log₂(8) + log₂(4)

Answer: 3 + 2 = 5

2. Simplify: log₅(125) − log₅(5)

Answer: 3 − 1 = 2

3. Solve for x: log₁₀(x) + log₁₀(x−3) = 1

Combine using product rule: log₁₀[x(x−3)] = 1

Convert: 10¹ = x(x−3)

10 = x² − 3x

x² − 3x − 10 = 0

(x−5)(x+2) = 0

x = 5 or x = −2

Reject −2 (log of negative doesn't exist). x = 5

4. Solve: 2^x = 100

Take ln: x · ln(2) = ln(100)

x = ln(100)/ln(2)

x ≈ 6.644

Where Logs Actually Appear

Logs aren't abstract math exercises. They show up in:

• Richter scale — earthquake magnitude uses log₁₀ of energy
• pH chemistry — negative log of hydrogen ion concentration
• Sound decibels — log₁₀ scale for intensity ratios
• Computer science — log time complexity in algorithms
• Finance — compound interest formulas use natural log for continuous growth

Understanding logs means understanding half the measurements in science and engineering.