Logarithm Examples- Practice Problems and Solutions

What You Need to Know About Logarithms

Logarithms trip up most students. The notation looks weird, the rules feel arbitrary, and the whole concept seems pointless until you realize exponential growth is everywhere — in finance, science, computer science, signal processing.

You can't avoid them. So let's get good at them.

Logarithm Basics: The Short Version

A logarithm answers one question: what exponent produces a given result?

If 2³ = 8, then log₂(8) = 3. That's it. The base is 2, the argument is 8, and the logarithm is the exponent.

Reading Logarithm Notation

log_b(x) = y means b^y = x

b = the base
x = the argument (what you're taking the log of)
y = the exponent you're solving for

Common Logarithm Bases

log with no subscript = base 10 (common logarithm)
ln = natural log, base e (approximately 2.718)
log₂ = base 2, common in computer science

The Core Logarithm Rules You Must Memorize

These four rules handle 95% of logarithm problems. Learn them until they're automatic.

1. Product Rule

log_b(MN) = log_b(M) + log_b(N)

When multiplying inside a log, you can split it into a sum of logs.

2. Quotient Rule

log_b(M/N) = log_b(M) - log_b(N)

When dividing inside a log, you get a difference of logs.

3. Power Rule

log_b(M^k) = k · log_b(M)

The exponent becomes a multiplier in front of the log.

4. Change of Base Formula

log_b(x) = log(x) / log(b) or log_b(x) = ln(x) / ln(b)

This lets you calculate logs in any base using a calculator that only has log₁₀ or ln.

Logarithm Examples: Worked Problems

Let's go through problems from simple to complex. Try each one before checking the solution.

Example 1: Basic Evaluation

Problem: Find log₂(32)

Solution:

Ask yourself: 2 to what power gives 32?

2⁵ = 32 ✓

Answer: 5

Example 2: Working with Variables

Problem: Solve for x: log₃(x) = 4

Solution:

Convert to exponential form: 3⁴ = x

3 · 3 · 3 · 3 = 81

Answer: x = 81

Example 3: Using the Product Rule

Problem: Simplify log₂(8 · 4)

Solution:

Apply the product rule: log₂(8) + log₂(4)

log₂(8) = 3 (since 2³ = 8)

log₂(4) = 2 (since 2² = 4)

Answer: 3 + 2 = 5

Example 4: Using the Quotient Rule

Problem: Simplify log₅(125/5)

Solution:

Apply the quotient rule: log₅(125) - log₅(5)

log₅(125) = 3 (since 5³ = 125)

log₅(5) = 1 (since 5¹ = 5)

Answer: 3 - 1 = 2

Example 5: Using the Power Rule

Problem: Simplify log₂(8³)

Solution:

Apply the power rule: 3 · log₂(8)

log₂(8) = 3

Answer: 3 · 3 = 9

Example 6: Solving a Log Equation

Problem: Solve for x: log₂(x) + log₂(x-2) = 3

Solution:

Combine using the product rule: log₂[x(x-2)] = 3

Convert to exponential: x(x-2) = 2³

x² - 2x = 8

x² - 2x - 8 = 0

Factor: (x-4)(x+2) = 0

x = 4 or x = -2

Check: x must be positive (log of negative doesn't exist in real numbers)

Answer: x = 4

Example 7: Change of Base

Problem: Calculate log₅(20) using a standard calculator

Solution:

Use change of base formula: log(20) / log(5)

Using a calculator: 1.3010 / 0.6990 ≈ 1.86

Answer: ≈ 1.86

Practice Problems: Your Turn

Work through these before checking the answers at the end.

log₁₀(1000) = ?
ln(e⁴) = ?
Solve for x: log₄(x) = 3
Simplify: log₃(9) + log₃(27)
Solve for x: 2^(log₂(x)) = 12

How to Solve Any Logarithm Problem: A Step-by-Step Process

When you see a log problem, follow this checklist:

Step 1: Identify What You're Solving For

Are you evaluating a log, simplifying an expression, or solving an equation? Different goals need different approaches.

Step 2: Check the Base

Is it base 10, base e, or something else? This affects how you might calculate the answer.

Step 3: Look for Patterns to Apply Rules

Do you see a product? Use the product rule. An exponent? Pull it out front. Multiple logs with the same base? Combine or separate them.

Step 4: Convert to Exponential Form If Stuck

log_b(x) = y becomes b^y = x. Sometimes working in exponential form is clearer.

Step 5: Check for Extraneous Solutions

Log arguments must be positive. If you solve and get x ≤ 0, that solution is invalid.

Log Rules Summary Table

Rule Name	Formula	When to Use
Product Rule	log_b(MN) = log_b(M) + log_b(N)	Multiplying inside a log
Quotient Rule	log_b(M/N) = log_b(M) - log_b(N)	Dividing inside a log
Power Rule	log_b(M^k) = k · log_b(M)	Exponent on the argument
Change of Base	log_b(x) = log(x)/log(b)	Calculating logs with any base
Log of 1	log_b(1) = 0	Any base, argument is 1
Log of Base	log_b(b) = 1	Argument equals the base

Common Mistakes That Cost You Points

Confusing log rules with exponent rules. log_b(M + N) ≠ log_b(M) + log_b(N). Only multiplication splits, not addition.
Forgetting the domain. log_b(x) requires x > 0. Always check your solutions.
Misapplying the power rule. log_b(M²) = 2 · log_b(M), not log_b(M²) = (log_b(M))².
Dropping the log entirely when solving equations without exponentiating both sides properly.

Answers to Practice Problems

3 (10³ = 1000)
4 (ln(e⁴) = 4 · ln(e) = 4 · 1 = 4)
x = 64 (4³ = 64)
4 (2 + 3 = 5... wait, log₃(9) = 2, log₃(27) = 3, so 2 + 3 = 5)
x = 12 (2^(log₂(x)) = x, so x = 12)

That's it. These examples cover the terrain. Practice the rules until you stop having to think about them.