Logarithmic Equations- Solving Methods and Examples

What Is a Logarithmic Equation?

A logarithmic equation contains a logarithm with an unknown variable inside it. The goal is always the same: find the value of that variable that makes the equation true.

These equations show up in science, engineering, finance, and computer science. If you're studying math past algebra, you'll need to solve them. Period.

The basic form looks like this:

log_b(x) = y

This means "b raised to what power gives you x?" The answer is y. That's it.

Core Logarithm Properties You Must Know

You can't solve these equations without the properties. Memorize them. I don't care how you do it—flashcards, repetition, tattoo them on your hand—but know these cold.

Product Rule

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

Quotient Rule

log_b(M/N) = log_b(M) − log_b(N)

Power Rule

log_b(M^p) = p · log_b(M)

Change of Base Formula

log_b(x) = log_a(x) / log_a(b)

This one is useful when you need to convert between bases or use a calculator that only has log₁₀.

How to Solve Logarithmic Equations

Here's the practical approach. Three main methods exist, and you pick based on what the equation looks like.

Method 1: Convert to Exponential Form

This works when you have a single logarithm on one side.

Steps:

• Identify the base
• Rewrite the logarithm as an exponential equation
• Solve the resulting equation
• Check your answer in the original equation

Method 2: Use Logarithm Properties First

When multiple logarithms appear, combine or simplify them using the rules above until you get a single log expression.

Method 3: Exponentiate Both Sides

When the equation has logs on both sides, raise both sides to the power of the base to eliminate the logarithms.

Solved Examples

Example 1: Basic Single Logarithm

Solve: log₃(x) = 4

Convert to exponential form:

3⁴ = x

3 × 3 × 3 × 3 = 81

Answer: x = 81

Check: log₃(81) = 4 ✓

Example 2: Logarithm with Addition

Solve: log₂(x) + log₂(x − 2) = 3

Use the product rule to combine:

log₂(x(x − 2)) = 3

log₂(x² − 2x) = 3

Convert to exponential:

x² − 2x = 2³

x² − 2x = 8

x² − 2x − 8 = 0

Factor:

(x − 4)(x + 2) = 0

x = 4 or x = −2

Check domain restrictions: The argument of a logarithm must be positive. x = −2 gives log₂(−2), which doesn't exist.

Answer: x = 4 only

Example 3: Natural Logarithm Equation

Solve: ln(x) + ln(x − 3) = ln(10)

Combine the left side using the product rule:

ln(x(x − 3)) = ln(10)

Since ln(a) = ln(b) means a = b:

x(x − 3) = 10

x² − 3x = 10

x² − 3x − 10 = 0

(x − 5)(x + 2) = 0

x = 5 or x = −2

Check: x = 5 works. x = −2 doesn't (negative argument for ln).

Answer: x = 5

Example 4: Logs on Both Sides

Solve: log₅(x + 1) = log₅(3x − 1)

When bases match, set arguments equal:

x + 1 = 3x − 1

2 = 2x

x = 1

Check: log₅(2) = log₅(2) ✓

Answer: x = 1

Example 5: Using Change of Base

Solve: log₂(x) = log₁₀(7)

Convert log₁₀(7) to base 2 using change of base:

log₁₀(7) = ln(7)/ln(2)

Or just use a calculator to get approximately 0.845:

log₂(x) ≈ 0.845

Convert to exponential:

x = 2^0.845

x ≈ 1.79

Answer: x ≈ 1.79

Common Mistakes That Will Sink You

• Ignoring domain restrictions: Logarithms only accept positive arguments. Always check that your solution makes the argument positive.
• Forgetting to check solutions: Extraneous solutions appear when you manipulate equations. Plug answers back into the original every time.
• Mixing up log rules: Product rule adds, quotient rule subtracts. Students mix these up constantly.
• Assuming both solutions work: In equations with multiple logs, often only one solution survives the domain check.

Quick Reference: Solving Methods

Practice Problems

Try these before checking answers:

1. log₄(x) = 3
2. log₂(x + 5) = 6
3. log(x) + log(x − 3) = 1
4. ln(x²) = 4
5. log₃(x) = log₃(2x − 9)

Answers:

1. x = 64
2. x = 59
3. x = 5 (x = −2 rejected)
4. x = ±e² ≈ ±7.39 (x = −e² rejected by domain)
5. x = 9 (x = 0 rejected)

The Bottom Line

Solving logarithmic equations comes down to three things: knowing your log properties, converting to exponential form when needed, and checking every solution against domain restrictions. That's it.

Don't overthink this. Practice twenty problems and you'll have it locked down.