How to Solve Equations When X Is an Exponent

Learn to solve exponential equations like 2^x = 16 or 3^2x+1 = 81 using logarithms and the same-base method.

📈 Algebra⏱️ ~10 min read📊 Intermediate

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What You'll Learn

✓When to use the "same base" method (fast, no calculator)
✓When and how to use logarithms
✓The power rule: log(a^x) = x·log(a)
✓How to handle messier problems with decimals

The Big Picture

In a normal equation like 3x = 12, the variable x is being multiplied—so you divide to solve. But in an exponential equation like 2^x = 16, the variable is in the exponent. You can't just divide; you need a different tool.

Two main approaches:

Same Base Method: If you can write both sides with the same base, just set the exponents equal.
Logarithm Method: When bases don't match nicely, take the log of both sides to "bring down" the exponent.

Method 1: Same Base (When Possible)

If both sides can be written as powers of the same base, the solution is quick.

Example: Solve 2^x = 16

Step 1: Rewrite 16 as a power of 2

16 = 2⁴

Step 2: Now the equation is 2^x = 2⁴

Step 3: Since bases match, set exponents equal

x = 4 ✓

Example: Solve 3^2x−1 = 81

Step 1: Rewrite 81 as a power of 3

81 = 3⁴

Step 2: Now 3^2x−1 = 3⁴

Step 3: Set exponents equal

2x − 1 = 4

Step 4: Solve for x

2x = 5

x = 5/2 = 2.5 ✓

Method 2: Using Logarithms

When you can't easily match bases (like solving 5^x = 17), logarithms are the tool.

Key Rule: Power Rule of Logarithms

log(a^x) = x · log(a)

Taking the log "brings down" the exponent as a multiplier.

Step-by-Step Process

Isolate the exponential expression on one side
Take log (or ln) of both sides
Use the power rule to bring the exponent down
Solve for x
Check with a calculator

Example: Solve 5^x = 17

Step 1: Take log of both sides

log(5^x) = log(17)

Step 2: Apply power rule on left side

x · log(5) = log(17)

Step 3: Solve for x

x = log(17) / log(5)

Step 4: Calculate

x ≈ 1.2304 / 0.6990 ≈ 1.760 ✓

Example: Solve 3^2x+1 = 50

Step 1: Take log of both sides

log(3^2x+1) = log(50)

Step 2: Power rule

(2x + 1) · log(3) = log(50)

Step 3: Divide both sides by log(3)

2x + 1 = log(50) / log(3)

2x + 1 ≈ 1.699 / 0.477 ≈ 3.561

Step 4: Solve for x

2x = 3.561 − 1 = 2.561

x ≈ 1.28 ✓

Which Method Should I Use?

✅ Use Same Base When:

• Both sides are powers of 2, 3, 5, 10, etc.
• Numbers like 4, 8, 16, 27, 81, 125
• You want an exact answer
• No calculator available

📊 Use Logarithms When:

• Bases don't match (like 5^x = 17)
• You need a decimal approximation
• The problem involves non-integer values
• Calculator is available

Common Mistakes to Avoid

⚠️

Confusing log(a+b) with log(a) + log(b)

Log of a sum is NOT the sum of logs. Only log(ab) = log(a) + log(b).

⚠️

Forgetting to apply log to BOTH sides

Whatever you do to one side, you must do to the other to keep the equation balanced.

⚠️

Not isolating the exponential first

If you have 2·5^x = 30, divide by 2 first to get 5^x = 15 before taking logs.

Quick Practice

Try these on your own:

Same Base:

4^x = 64
2^3x−2 = 32
9^x = 27 (hint: both are powers of 3)

Logarithms:

7^x = 20
4^x+1 = 100
2^x = 3^x−1

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