Solving Exponential Equations- Methods and Examples

What Is an Exponential Equation?

An exponential equation has a variable in the exponent. That's it. The base can be any number, but the defining feature is that x sits up top where exponents live.

Examples:

• 2^x = 8
• 5^3x-2 = 125
• e^x = 10
• 3^x+1 = 27

If you see a variable in the exponent, you're dealing with an exponential equation. Now let's solve them.

Method 1: Match the Bases

This is the easiest method when it works. Rewrite both sides so they have the same base, then set the exponents equal.

Example

Solve: 2^x = 8

8 is 2³. Rewrite:

2^x = 2³

Now x = 3. Done.

Example

Solve: 5^3x-2 = 125

125 is 5³. Rewrite:

5^3x-2 = 5³

3x - 2 = 3

3x = 5

x = 5/3

This method works great when you can quickly identify the base. Not always possible, though. That's where logarithms come in.

Method 2: Use Logarithms

When bases won't match, take the log of both sides. Any base works—common log (log), natural log (ln), it doesn't matter.

Example

Solve: 3^x = 20

Take log of both sides:

log(3^x) = log(20)

Use the power rule: x · log(3) = log(20)

x = log(20) / log(3)

x ≈ 2.73

Example

Solve: e^x = 10

Use ln since the base is e:

ln(e^x) = ln(10)

x · ln(e) = ln(10)

x = ln(10)

x ≈ 2.303

Method 3: Take the Log of Both Sides Directly

For equations like a^x = b^y where bases differ, take log of both sides immediately.

Example

Solve: 2^x = 3⁵

log(2^x) = log(3⁵)

x · log(2) = 5 · log(3)

x = 5 · log(3) / log(2)

x ≈ 7.92

Method 4: Substitution

When you have the same base raised to different expressions, substitution helps isolate things first.

Example

Solve: 4^x - 5 · 2^x + 4 = 0

Notice 4^x = (2²)^x = 2^2x = (2^x)²

Let u = 2^x

Then: u² - 5u + 4 = 0

Factor: (u - 1)(u - 4) = 0

u = 1 or u = 4

Back-substitute:

2^x = 1 → x = 0

2^x = 4 → x = 2

Solutions: x = 0, x = 2

Comparing the Methods

How to Get Started: Step-by-Step

When you see an exponential equation, follow this decision path:

1. Can you rewrite the non-variable side as a power of the base? → If yes, match bases.
2. Is the variable in the exponent only? → Take log of both sides.
3. Are there multiple terms with the same base? → Try substitution (factor into a quadratic).
4. Still stuck? → Take log of both sides first, then isolate using algebra.

Common Mistakes to Avoid

• Forgetting to take the log of both sides — you can't solve 3^x = 20 by dividing. That's algebra, not exponentials.
• Applying log to only part of an equation — whatever you do to one side, you must do to the other.
• Confusing log rules — log(aⁿ) = n · log(a), not n + log(a).
• Ignoring domain restrictions — logs require positive arguments.

Practice Problems

1. Solve: 7^x = 343

Answer: x = 3 (343 = 7³)

2. Solve: 4^2x+1 = 64

Answer: x = 1 (64 = 4³, so 2x+1 = 3)

3. Solve: 2^x = 50

Answer: x = log(50)/log(2) ≈ 5.64

4. Solve: 9^x - 6 · 3^x - 27 = 0

Answer: x = 3 (use u = 3^x, get u² - 6u - 27 = 0)

Bottom Line

Exponential equations aren't hard once you know your options. Match bases when it's easy. Use logs when it isn't. Substitute when things look quadratic. That's the whole game.