How Do Logarithms Work? Understanding Log Basics

What Is a Logarithm, Really?

A logarithm is just the inverse of an exponent. That's it. If 2³ = 8, then log₂(8) = 3. The log asks: "What power do I need to raise this base to, to get that number?"

Most people panic at the sight of log notation. Don't. It's simpler than algebra class made it seem. You're just solving for the unknown exponent.

The Core Relationship

Logs and exponents are two sides of the same coin. Here's the swap:

Exponent form: base^exponent = answer

Log form: log_base(answer) = exponent

When you see log_b(x) = y, it means b^y = x. That's the whole translation.

A Quick Example

log₁₀(1000) = 3

This asks: What power of 10 gives you 1000? The answer is 3, because 10³ = 1000.

Another One

log₂(32) = 5

Because 2⁵ = 32. Easy.

Common Log Bases You Need to Know

Most logs use one of three bases. Learn these and you're set.

Base 10 (Common Log)

Written as log(x) without a subscript. This is the default in most textbooks and calculators.

log(100) = 2 because 10² = 100.

Base e (Natural Log)

Written as ln(x). The "e" is approximately 2.718. Natural logs show up constantly in science and finance because they describe natural growth and decay.

ln(e) = 1. By definition.

Base 2 (Binary Log)

Written as log₂(x). This is huge in computer science. Binary means 2, and it's everywhere in computing.

log₂(8) = 3. log₂(1024) = 10.

Why Logs Even Exist

Logs solve a specific problem: they let you work with huge numbers without writing them out. Instead of multiplying millions, you add logs.

Before calculators, engineers used log tables to multiply massive numbers by hand. The math was slow but doable.

Today, logs show up in:

• Measuring earthquake strength (Richter scale uses base 10 logs)
• Sound loudness (decibels)
• Compound interest calculations
• Algorithm complexity in computer science
• Signal processing and data compression

Any time you compress a massive range into something manageable, you're probably using logs.

Log Rules (The Short List)

You need these to work with logs without pulling out a calculator for everything.

• Product rule: log(ab) = log(a) + log(b)
• Quotient rule: log(a/b) = log(a) - log(b)
• Power rule: log(aⁿ) = n · log(a)
• Change of base: logₐ(x) = log(x) / log(a)

The change of base formula is the practical one. It lets you calculate logs in any base using a calculator that only does base 10 or natural logs.

Common Logarithm Types Compared

Getting Started: How to Actually Use Logs

Step 1: Identify the Base

Look for the small subscript number. No subscript usually means base 10. "ln" always means natural log.

Step 2: Ask the Right Question

log₂(64) = ? → "2 to what power gives 64?"

Step 3: Solve or Estimate

2⁶ = 64, so log₂(64) = 6.

If you can't solve it exactly, use the change of base formula:

log₂(50) = log(50) / log(2) ≈ 1.699 / 0.301 ≈ 5.64

Check: 2⁵·⁶⁴ ≈ 50. It works.

Step 4: Apply the Rules When Needed

Multiply numbers? Add their logs. Divide? Subtract logs. Raise to a power? Multiply the log by the exponent.

Example: 5³ × 5⁴ = 5⁷

In logs: log(5³) + log(5⁴) = 3·log(5) + 4·log(5) = 7·log(5) = log(5⁷)

What About Logarithms of Special Numbers?

• log(1) = 0 always. Any base to the power of 0 equals 1.
• log(base) = 1 always. Any base to the first power equals itself.
• log(negative) = undefined in real numbers. You can't raise a positive base to get a negative result.
• log(0) = undefined. No exponent ever produces 0.

When Logs Get Weird: Domains and Restrictions

Logs only accept positive inputs. The argument (the number inside the parentheses) must be greater than 0.

log(-4) doesn't exist in the real number system. Neither does log(0). If your problem asks for these, the answer is "undefined" or "no solution."

This trips people up constantly. Don't get caught.

Natural Log vs. Common Log: When to Use Which

Use common log (log) when:

• Working in general math classes
• No specific base is required
• Your calculator only has "log" and "ln"

Use natural log (ln) when:

• Calculus problems
• Modeling growth, decay, or anything continuous
• Finance, biology, physics formulas

The math works out cleaner with ln in advanced contexts. That's why it became the default in higher math.

Real Example: Earthquake Magnitude

The Richter scale is logarithmic. A magnitude 6 earthquake isn't twice as strong as magnitude 3—it's 1,000 times stronger.

Magnitude difference of 1 = factor of 10 in ground motion.

Magnitude difference of 2 = factor of 100.

Magnitude difference of 3 = factor of 1,000.

That's what logs do. They compress a massive range into readable numbers.

Quick Reference: Common Log Values

• log₁₀(1) = 0
• log₁₀(10) = 1
• log₁₀(100) = 2
• log₁₀(1000) = 3
• log₁₀(10,000) = 4
• ln(e) = 1
• ln(1) = 0
• log₂(8) = 3
• log₂(1024) = 10

Memorize these. They'll save you time on tests and in calculations.