What's the difference between \log_a x and \ln x?

The natural logarithm \ln x is a logarithm with base e (approximately 2.718), and follows the same definition: \ln x = y means e^y = x. Natural logarithms are widely used in science and calculus because e has special mathematical properties.

Why can't I take the logarithm of a negative number?

By the definition of logarithm, \log_a x = y means a^y = x. Since a > 0, any power a^y is always positive, so x cannot be negative or zero. The argument must always be positive for the logarithm to be defined.

How do I know if my logarithm answer is right?

Use the definition to check: if you found \log_a x = y, verify that a^y = x. For example, if you solved \log_3 27 = y and got y = 3, check that 3^3 = 27, which is true.

When would I actually use logarithms in real life?

Logarithms appear in science, engineering, and finance whenever something grows or shrinks exponentially. For example, if money grows at 5% per year, you use logarithms to find how long it takes to double; if radioactive material decays, you use logarithms to find its half-life.

Definition of a Logarithm

Use the definition of a logarithm to convert between exponential and logarithmic form when solving for unknown exponents.

\log_a x = y \iff a^y = x

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What each symbol means

What Definition of a Logarithm takes

Symbol	Meaning
$a$	The base—the number being raised to a power; must be positive and not equal to 1, or the logarithm is undefined.
$x$	The argument—the positive result you get after raising the base to some power; must be positive, or the logarithm is undefined.
$y$	The exponent or output of the logarithm—the power you raise the base to; can be any real number (positive, negative, or zero).

When to use it

When you need to solve for an exponent in an equation and recognize that conversion between exponential and logarithmic form reveals the answer.

Level

Usually taught in: Algebra II

Worked examples

1. Find the logarithm with integer result

Problem

Solve

\log_3 27 = y

for

y

$3^y = 27$
By the definition of logarithm, $\log_3 27 = y$ means we need the value of $y$ such that $3^y = 27$ .
$27 = 3^3$
Recognize that $27 = 3^3$ , so we can rewrite the equation.
$3^y = 3^3$
Substitute this into the equation from step 1.
$y = 3$
When the bases are equal, the exponents must be equal.

Answer: $y = 3$

The definition of logarithm instantly converts the question into finding which power of 3 gives 27. Recognizing that $27 = 3^3$ makes the exponent obvious by equating exponents on both sides.

2. Solve a logarithm equation with a negative result

Problem

Solve

\log_5 \frac{1}{25} = y

for

y

$5^y = \frac{1}{25}$
Apply the definition of logarithm to convert $\log_5 \frac{1}{25} = y$ into exponential form.
$\frac{1}{25} = \frac{1}{5^2} = 5^{-2}$
Rewrite the fraction using a negative exponent: $5^{-2}$ means $\frac{1}{5^2}$ .
$5^y = 5^{-2}$
Substitute this expression back into the equation from step 1.
$y = -2$
Equate the exponents since the bases are equal.

Answer: $y = -2$

When the argument of a logarithm is a fraction less than 1, the result is negative. Converting to exponential form and rewriting the fraction as a negative power of the base makes it clear that $y = -2$ .

3. Find the time for an investment to double using logarithms

Problem

An investment grows according to

A = 500 \cdot 1.08^t

, where

A

is the amount in dollars and

t

is the number of years. How many years will it take for the account to reach $1000?

$1000 = 500 \cdot 1.08^t$
Set up the equation by substituting $A = 1000$ into the growth formula.
$2 = 1.08^t$
Divide both sides by 500 to isolate the exponential term.
$\log_{1.08} 2 = t$
By the definition of logarithm, $\log_{1.08} 2 = t$ is equivalent to $1.08^t = 2$ .

Answer: $t = \log_{1.08} 2 \approx 9.0 \text{ years}$

The definition of logarithm converts the exponential equation into a form that directly identifies the exponent $t$ . Using a calculator or the change of base formula, $\log_{1.08} 2 \approx 9.0$ years.

Common mistakes

Where Definition of a Logarithm usually goes wrong

The mistake	Why it is wrong	The fix
Writing $\log_a x = y$ and then computing $a \cdot x = y$	The definition uses exponentiation, not multiplication; the logarithm and exponentiation are inverse operations.	Always convert using the definition: $\log_a x = y$ means $a^y = x$ , where $a$ is the base being raised to power $y$ .
Confusing which number is the base and which is the exponent, writing $\log_3 81 = y$ as $y^3 = 81$ instead of $3^y = 81$	The base of the logarithm appears as the base of the exponential form, not as the exponent; what you are solving for ( $y$ ) becomes the exponent in the exponential form.	The base $a$ goes in the exponent form as the base: $a^y = x$ . The result $y$ (what the logarithm equals) becomes the exponent, never the base.
Trying to take the logarithm of a negative number or zero, like $\log_2(-4)$ or $\log_3 0$	Logarithms are undefined for non-positive arguments because no real exponent $y$ exists such that $a^y \leq 0$ when $a > 0$ and $a \neq 1$ .	Always verify that the argument (the number inside the logarithm) is positive before applying the logarithm definition.

Tips and when to use something else

Always convert $\log_a x = y$ to its exponential form $a^y = x$ to check your answer or solve for the unknown.
If the base of the logarithm is not an obvious power of the argument, use the change of base formula to evaluate it: $\log_a x = \frac{\log x}{\log a}$ .
Watch the domain: the base must be positive and not 1, and the argument must be positive; otherwise the logarithm is undefined.
For solving exponential equations like $2^x = 7$ where you cannot easily spot the exponent, convert to logarithmic form using the definition: $x = \log_2 7$ .

Related formulas

Change of Base FormulaConvert logarithms between different bases so you can evaluate them on a calculator or combine logarithms with different bases.Product Rule for LogarithmsCombine multiplication inside a logarithm into a sum of separate logarithms, making complex equations and expressions simpler to solve and evaluate.Natural LogarithmNatural logarithm finds the power to which e must be raised to get a number, and it's essential for solving equations involving exponential functions.Exponential GrowthUse Exponential Growth when a quantity grows by multiplying by a constant factor each time period, like bacteria populations or money earning compound interest.

Frequently asked questions

What's the difference between $\log_a x$ and $\ln x$ ?: The natural logarithm $\ln x$ is a logarithm with base $e$ (approximately 2.718), and follows the same definition: $\ln x = y$ means $e^y = x$ . Natural logarithms are widely used in science and calculus because $e$ has special mathematical properties.
Why can't I take the logarithm of a negative number?: By the definition of logarithm, $\log_a x = y$ means $a^y = x$ . Since $a > 0$ , any power $a^y$ is always positive, so $x$ cannot be negative or zero. The argument must always be positive for the logarithm to be defined.
How do I know if my logarithm answer is right?: Use the definition to check: if you found $\log_a x = y$ , verify that $a^y = x$ . For example, if you solved $\log_3 27 = y$ and got $y = 3$ , check that $3^3 = 27$ , which is true.
When would I actually use logarithms in real life?: Logarithms appear in science, engineering, and finance whenever something grows or shrinks exponentially. For example, if money grows at 5% per year, you use logarithms to find how long it takes to double; if radioactive material decays, you use logarithms to find its half-life.

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Reviewed 2026-09-18