Lesson 1: Logarithmic Functions

New Concept

Today, we'll explore logarithmic functions, the powerful inverses of exponential functions. You'll learn how this inverse relationship, $y = \operatorname{log}_b x \iff x = b^y$, helps us graph, evaluate, and solve a new class of equations.

What’s next

Let's start by exploring this inverse relationship. You will work through interactive examples and practice cards to master graphing and evaluating these new functions.

Property

Two functions are inverse functions if each one undoes the effect of the other. The graphs of inverse functions are symmetric about the line $y = x$. If we interchange the variables in the function, we get an equivalent formula for its inverse. For example, $y = \sqrt[3]{x}$ if and only if $x = y^3$.

Examples

• The inverse of taking the fifth power of a number is taking the fifth root. If we start with $x=2$, taking the fifth power gives $2^5=32$. The fifth root of 32 is $\sqrt[5]{32}=2$, our original number.

• The functions $f(x) = x+7$ and $g(x) = x-7$ are inverses. If you take a number, say 20, then $f(20) = 27$. Applying the inverse gives $g(27) = 20$, returning to the start.

Property

The logarithmic function is the inverse of the exponential function. Each function undoes the effect of the other. The relationship is defined as: $y = \log_b x \text{ if and only if } x = b^y$. Because they are inverse functions, these two identities always hold for $b>0$:

Examples

• To simplify $\log_3(3^5)$, notice the logarithm base 3 undoes the exponential base 3. The expression simplifies to the exponent, which is 5.

• To evaluate $10^{\log_{10} 100}$, the exponential base 10 undoes the logarithm base 10. The expression simplifies to 100.

Property

The graph of a logarithmic function $g(x) = \log_b x$ can be found by reflecting the graph of its inverse exponential function, $f(x) = b^x$, across the line $y=x$. To create a table of values for $g(x) = \log_b x$, you can interchange the columns in a table for $f(x) = b^x$.

Examples

• Since the point $(3, 27)$ is on the graph of the exponential function $y=3^x$, the point $(27, 3)$ must be on the graph of the logarithmic function $y=\log_3 x$.

• The graph of $y=10^x$ passes through $(0, 1)$ and $(1, 10)$. Therefore, the graph of its inverse, $y=\log_{10} x$, must pass through $(1, 0)$ and $(10, 1)$.

Property

For any base $b > 0$, $b \neq 1$:

1. The logarithmic function $y = \log_b x$ is defined for positive $x$ only.
2. The $x$-intercept of its graph is $(1, 0)$.
3. The graph has a vertical asymptote at $x = 0$.
4. The graphs of $y = \log_b x$ and $y = b^x$ are symmetric about the line $y = x$.

Examples

• The expression $\log_5(-25)$ is undefined because the input to a logarithm must be a positive number. You cannot take the log of a negative number or zero.

• For any valid base $b$, the value of $\log_b 1$ is always 0. For instance, $\log_{20}(1) = 0$. This gives the graph its characteristic x-intercept at $(1, 0)$.

Property

To solve a logarithmic equation:

1. If the equation contains only one logarithm, convert it to its equivalent exponential form using the definition $y = \log_b x \iff x = b^y$.
2. If the equation contains more than one logarithm, use logarithm properties to combine them into a single logarithm first.
3. Always check for extraneous solutions, because the argument of a logarithm must be positive.

Examples

• To solve $\log_5(x+3) = 2$, convert it to exponential form: $x+3 = 5^2$. This gives $x+3=25$, so $x=22$. The solution is valid.

• Solve $\log_3 x + \log_3(x-6) = 3$. First combine the logs: $\log_3(x(x-6)) = 3$. Convert to exponential form: $x(x-6) = 3^3$, which is $x^2-6x=27$. The solutions to $x^2-6x-27=0$ are $x=9$ and $x=-3$. Checking them, $x=9$ is valid, but $x=-3$ is extraneous because $\log_3(-3)$ is undefined.