Lesson 6.3 : Logarithmic Functions

New Concept

A logarithm is the exponent needed to raise a base to get a certain number. We'll explore this inverse relationship to exponents by converting between forms ($y = \operatorname{log}_b(x) \iff b^y = x$) and evaluating common and natural logs.

What’s next

Get ready to master these concepts! You'll soon work through interactive examples, a series of practice cards, and challenge problems to solidify your understanding.

Property

A logarithm base $b$ of a positive number $x$ satisfies the following definition. For $x > 0$, $b > 0$, $b \neq 1$,

Examples

• The logarithmic equation $\operatorname{log}_7(\sqrt{7}) = \frac{1}{2}$ is equivalent to the exponential equation $7^{\frac{1}{2}} = \sqrt{7}$.

• The equation $\operatorname{log}_5(25) = 2$ can be rewritten in exponential form as $5^2 = 25$.

Property

To convert from exponents to logarithms, we follow the same steps in reverse. For an exponential equation $b^y = x$, identify the base $b$, exponent $y$, and output $x$. Then rewrite the equation in logarithmic form as $\operatorname{log}_b(x) = y$. This process switches the roles of the input and output, reflecting that logarithms are the inverse of exponential functions.

Examples

• The exponential equation $3^4 = 81$ is equivalent to the logarithmic equation $\operatorname{log}_3(81) = 4$.

• The equation $6^2 = 36$ can be rewritten in logarithmic form as $\operatorname{log}_6(36) = 2$.

Property

Given a logarithm of the form $y = \operatorname{log}_b(x)$, evaluate it mentally by following these steps:

1. Rewrite the argument $x$ as a power of $b$: $b^y = x$.
2. Use previous knowledge of powers of $b$ to identify $y$ by asking, “To what exponent should $b$ be raised in order to get $x$?”

Examples

• To solve $y = \operatorname{log}_5(125)$, rewrite it as $5^y = 125$. Since we know $5^3 = 125$, the solution is $y=3$.

• To evaluate $\operatorname{log}_2(\frac{1}{32})$, ask 'To what power must 2 be raised to get $\frac{1}{32}$?' Since $2^5 = 32$, we know $2^{-5} = \frac{1}{32}$. The answer is -5.

Property

A common logarithm is a logarithm with base 10. We write $\operatorname{log}_{10}(x)$ simply as $\operatorname{log}(x)$. The common logarithm of a positive number $x$ satisfies the following definition.
For $x > 0$,

Examples

• To evaluate $\operatorname{log}(10000)$, you are solving $\operatorname{log}_{10}(10000)$. Since $10^4 = 10000$, the answer is 4.

• To evaluate $\operatorname{log}(0.001)$, you are solving $\operatorname{log}_{10}(\frac{1}{1000})$. Since $10^{-3} = \frac{1}{1000}$, the answer is -3.

Property

A natural logarithm is a logarithm with base $e$. We write $\operatorname{log}_e(x)$ simply as $\ln(x)$. The natural logarithm of a positive number $x$ satisfies the following definition.
For $x > 0$,

Examples

• To evaluate $\ln(e^5)$, remember that natural log and $e$ are inverses. The function cancels the base, leaving just the exponent. The answer is 5.

• To evaluate $\ln(\frac{1}{e^2})$, first rewrite the expression as $\ln(e^{-2})$. The $\ln$ and $e$ cancel each other out, so the result is -2.