Lesson 10.3: Evaluate and Graph Logarithmic Functions

New Concept

Logarithmic functions are the inverses of exponential functions, meaning $y = \log_a x$ is equivalent to $x = a^y$. This core relationship is the key to converting between forms, evaluating expressions, graphing functions, and solving logarithmic equations.

What’s next

First, you'll practice converting between logarithmic and exponential forms with interactive examples. Then, you'll apply this skill to graphing functions and solving equations in practice cards.

Property

The function $f(x) = \log_a x$ is the logarithmic function with base $a$, where $a > 0, x > 0$, and $a \neq 1$.

$y = \log_a x$ is equivalent to $x = a^y$

To help with converting, remember: “base to the exponent gives us the number.” The logarithm is the exponent.

Property

We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation. To find the exact value of a logarithm like $\log_b N$, you can set the expression equal to $x$ (so $\log_b N = x$) and then convert it into the exponential equation $b^x = N$ to solve for $x$.

Examples

• To find the value of $x$ in $\log_x 81 = 2$, convert to exponential form: $x^2 = 81$. Since the base must be positive, $x=9$.
• To find the value of $x$ in $\log_2 x = 5$, convert to exponential form: $2^5 = x$. Therefore, $x=32$.
• To find the exact value of $\log_5 \frac{1}{25}$, set it to $x$. $\log_5 \frac{1}{25} = x$ becomes $5^x = \frac{1}{25}$. Since $\frac{1}{25} = 5^{-2}$, we find $x = -2$.

Explanation

Evaluating a logarithm means finding the exponent. By asking, 'What power do I raise the base to get the number?', you are setting up the problem. Converting to an exponential equation is the formal way to solve for that unknown exponent.

Property

Properties of the Graph of $y = \log_a x$

When $a > 1$

• Domain: $(0, \infty)$
• Range: $(-\infty, \infty)$
• $x$-intercept: $(1, 0)$
• Asymptote: $y$-axis
• Basic shape: increasing

When $0 < a < 1$

• Domain: $(0, \infty)$
• Range: $(-\infty, \infty)$
• $x$-intercept: $(1, 0)$
• Asymptote: $y$-axis
• Basic shape: decreasing

Property

The function $f(x) = \ln x$ is the natural logarithmic function with base $e$, where $x > 0$. $y = \ln x$ is equivalent to $x = e^y$.
The function $f(x) = \log x$ is the common logarithmic function with base 10, where $x > 0$. $y = \log x$ is equivalent to $x = 10^y$.
To solve logarithmic equations, change the equation to exponential form and solve. Always check solutions to eliminate any that make the logarithm's argument non-positive.

Examples

• To solve $\log_a 121 = 2$, rewrite in exponential form as $a^2 = 121$. Since the base $a$ must be positive, $a=11$.
• To solve $\ln x = 4$, rewrite in exponential form using base $e$. This gives $x = e^4$.
• To solve $\log_4(2x+1) = 2$, rewrite as $4^2 = 2x+1$. This simplifies to $16 = 2x+1$, so $15 = 2x$, and $x = 7.5$.

Explanation

Natural log (ln) uses base $e$ and common log (log) uses base 10. The most direct way to solve a logarithmic equation is to rewrite it in its equivalent exponential form. Always check your final answer in the original equation.

Property

The loudness level, $D$, measured in decibels, of a sound of intensity, $I$, measured in watts per square inch is

Examples

• The decibel level of a quiet room with intensity $10^{-10}$ watts per square inch is $D = 10 \log(\frac{10^{-10}}{10^{-12}}) = 10 \log(10^2) = 10 \cdot 2 = 20$ dB.
• The sound of a leaf blower with intensity $10^{-1}$ watts per square inch is $D = 10 \log(\frac{10^{-1}}{10^{-12}}) = 10 \log(10^{11}) = 10 \cdot 11 = 110$ dB.
• Normal conversation with intensity $10^{-6}$ watts per square inch is $D = 10 \log(\frac{10^{-6}}{10^{-12}}) = 10 \log(10^6) = 10 \cdot 6 = 60$ dB.

Explanation

This formula compares a sound's intensity ($I$) to the quietest sound humans can hear ($10^{-12}$). The logarithm converts the vast range of sound intensities into a more manageable scale, typically from 0 to 160 decibels (dB).

Property

The magnitude $R$ of an earthquake is measured by $R = \operatorname{log} I$, where $I$ is the intensity of its shock wave. This is known as the Richter scale.

Examples

• An earthquake with a magnitude of $R=5$ on the Richter scale has an intensity found by solving $5 = \log I$. In exponential form, this is $I = 10^5$.
• Compare a magnitude 8 earthquake to a magnitude 6 one. Their intensities are $I_8 = 10^8$ and $I_6 = 10^6$. The ratio $\frac{10^8}{10^6} = 10^2 = 100$ means the magnitude 8 quake is 100 times more intense.
• The 1989 Loma Prieta earthquake had a magnitude of 6.9, so its intensity was $I = 10^{6.9}$. A smaller tremor of magnitude 4.9 has an intensity of $I = 10^{4.9}$. The larger quake was $\frac{10^{6.9}}{10^{4.9}} = 10^2 = 100$ times more intense.

Explanation

The Richter scale uses a common logarithm (base 10) to measure an earthquake's intensity, $I$. Each whole number increase on this scale represents a tenfold increase in the measured intensity of the ground shaking.