How do I know which base to use for the logarithm?

Use the same base as the base of the exponential term. If your equation is 2^x = 50, take \log_2 of both sides. The base of the logarithm must match the base of the exponential so that the property \log_a(a^x) = x applies correctly.

Can I have a negative or zero base in an exponential equation?

No. The base a must be positive and not equal to 1; otherwise the logarithm is undefined. Also, b must be positive because you cannot take the logarithm of a negative number in real-number arithmetic.

What if I can't evaluate the logarithm by hand, like in 2^x = 50?

Use the change of base formula to convert to a common or natural logarithm: x = \log_2(50) = \frac{\log(50)}{\log(2)} \approx 5.64. This allows you to use a calculator to find a decimal approximation of the answer.

What's the difference between an exponential equation and a logarithmic equation?

An exponential equation has the unknown in the exponent, like 2^x = 8. A logarithmic equation has the unknown inside a logarithm, like \log_2(x) = 3. They are inverses: solving 2^x = 8 with logarithms gives x = \log_2(8), which is a logarithmic statement.

Exponential Equations

Solve exponential equations where an unknown exponent needs to be found using logarithms to transform them into solvable linear form.

a^x = b \implies x = \log_a b

Solve a problem with Exponential Equations

Type the problem. The solver will use Exponential Equations where Exponential Equations is the right tool, and tell you when it is not.

What each symbol means

What Exponential Equations takes

Symbol	Meaning
$a$	The base of the exponential expression; must be positive and not equal to 1, or the logarithm is undefined.
$b$	The result value on the right side of the equation; must be positive, as you cannot take the logarithm of a negative number.
$x$	The unknown exponent you are solving for; can be any real number, including negative or fractional values.

When to use it

Use exponential equations when you need to find an unknown exponent in a power equation.

Level

Usually taught in: Algebra II

Worked examples

1. Solve a simple exponential equation with integer base and result

Problem

Solve

2^x = 16

for

x

$2^x = 16$
This is the original exponential equation where the unknown exponent $x$ is what we need to find.
$\log_2(2^x) = \log_2(16)$
Take $\log_2$ (logarithm base 2) of both sides to bring down the exponent.
$x = \log_2(16)$
Apply the property $\log_a(a^x) = x$ , which says the base- $a$ logarithm of $a^x$ is just $x$ .
$x = 4$
Evaluate $\log_2(16)$ by recognizing that $2^4 = 16$ , so $\log_2(16) = 4$ .

Answer: $x = 4$

We converted the exponential equation into logarithmic form to isolate $x$ . Since $2^4 = 16$ , the answer checks: $2^4 = 16$ ✓.

2. Solve an exponential equation with a fractional result

Problem

Solve

3^x = \frac{1}{9}

for

x

$3^x = \frac{1}{9}$
This equation has a fractional result less than 1, which suggests $x$ will be negative since $3^{-n} = \frac{1}{3^n}$ .
$\log_3(3^x) = \log_3\left(\frac{1}{9}\right)$
Take $\log_3$ of both sides to isolate the exponent.
$x = \log_3\left(\frac{1}{9}\right)$
Apply $\log_a(a^x) = x$ on the left side, leaving $x$ by itself.
$x = -2$
Evaluate by recognizing that $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$ , so $\log_3\left(\frac{1}{9}\right) = -2$ .

Answer: $x = -2$

Fractional results in exponential equations lead to negative exponents. The negative answer makes sense: $3^{-2} = \frac{1}{9}$ ✓.

3. Solve a word problem with exponential growth

Problem

A basketball player's scoring per game increases exponentially according to the formula

P = 10 \cdot 2^n

, where

P

is the points scored and

n

is the game number. In which game will the player score 320 points?

$10 \cdot 2^n = 320$
Set up the equation by substituting $P = 320$ into the scoring formula.
$2^n = 32$
Divide both sides by 10 to isolate the exponential term.
$\log_2(2^n) = \log_2(32)$
Take $\log_2$ of both sides to bring down the exponent $n$ .
$n = \log_2(32)$
Apply the property $\log_a(a^x) = x$ to simplify the left side.
$n = 5$
Evaluate $\log_2(32)$ by recognizing that $2^5 = 32$ .

Answer: $n = 5$

The player will score 320 points in game 5. We can verify: $P = 10 \cdot 2^5 = 10 \cdot 32 = 320$ ✓.

Common mistakes

Where Exponential Equations usually goes wrong

The mistake	Why it is wrong	The fix
Taking the logarithm of only one side, writing $\log(2^x) = 8$ instead of $\log(2^x) = \log(8)$ .	Logarithm is a function, and to preserve equality you must apply it to both sides of the equation, just like with any algebraic operation.	Always take the logarithm of both sides: if $a^x = b$ , write $\log_a(a^x) = \log_a(b)$ , then simplify.
Forgetting to apply the property $\log_a(a^x) = x$ after taking the logarithm, leaving you with $\log_3(8)$ and not knowing what to do next.	The logarithm property is what actually isolates $x$ and makes the problem solvable; skipping it leaves $x$ trapped inside a logarithm.	After taking the logarithm of both sides, immediately simplify $\log_a(a^x)$ to $x$ before attempting to evaluate the right side.
Using the wrong base for the logarithm, such as taking $\log_{10}(3^x) = \log_{10}(8)$ instead of $\log_3(3^x) = \log_3(8)$ .	The logarithm property $\log_a(a^x) = x$ only works when the base of the logarithm matches the base of the exponential.	Match the logarithm base to the exponential base: for $a^x = b$ , use $\log_a$ on both sides, not a different base.

Tips and when to use something else

If both sides of the equation already have the same base, such as $2^x = 2^5$ , you can skip logarithms and equate the exponents directly to get $x = 5$ .
For equations like $2^x = 8$ , try to rewrite the right side as a power of the left base (for example, $8 = 2^3$ ); this makes solving possible without a calculator.
When bases do not match and you need an exact answer, the solution is a logarithm; use Logarithmic Equations if your problem asks you to simplify $x = \log_a(b)$ further or evaluate it numerically.
Exponential equations model real-world situations like population growth, radioactive decay, bacterial growth, and compound interest; recognizing when a word problem leads to an exponential equation is key to setting it up correctly.

Related formulas

Logarithmic EquationsLearn to solve logarithmic equations by converting to exponential form, the key skill for finding unknown values hidden inside logarithms.Extraneous SolutionsA solution to the transformed equation that doesn't satisfy the original; found when squaring both sides or applying other non-reversible operations.Compound InterestCompound Interest calculates how money grows when interest earns interest, used for investments, savings, and loans over time.Scientific NotationScientific notation expresses very large or very small numbers as a coefficient times a power of ten, making them easier to work with.

Frequently asked questions

How do I know which base to use for the logarithm?: Use the same base as the base of the exponential term. If your equation is $2^x = 50$ , take $\log_2$ of both sides. The base of the logarithm must match the base of the exponential so that the property $\log_a(a^x) = x$ applies correctly.
Can I have a negative or zero base in an exponential equation?: No. The base $a$ must be positive and not equal to 1; otherwise the logarithm is undefined. Also, $b$ must be positive because you cannot take the logarithm of a negative number in real-number arithmetic.
What if I can't evaluate the logarithm by hand, like in $2^x = 50$ ?: Use the change of base formula to convert to a common or natural logarithm: $x = \log_2(50) = \frac{\log(50)}{\log(2)} \approx 5.64$ . This allows you to use a calculator to find a decimal approximation of the answer.
What's the difference between an exponential equation and a logarithmic equation?: An exponential equation has the unknown in the exponent, like $2^x = 8$ . A logarithmic equation has the unknown inside a logarithm, like $\log_2(x) = 3$ . They are inverses: solving $2^x = 8$ with logarithms gives $x = \log_2(8)$ , which is a logarithmic statement.

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Reviewed 2026-09-18