Lesson 4: Chapter Summary and Review

New Concept

This lesson explores the powerful relationship between exponential and logarithmic functions. You'll learn how they are inverses of each other, allowing you to solve for variables in exponents and apply these concepts to real-world growth and decay models.

What’s next

Next, you'll tackle interactive examples on solving logarithmic equations and apply your skills to real-world problems involving exponential growth and decay.

Property

Two functions are called inverse functions if each function undoes the effects of the other. If we apply the inverse function to the output of $f$, we return to the original input value. The graphs of $f$ and its inverse function are symmetric about the line $y = x$.

Examples

• If $f(x) = x + 4$, its inverse is $f^{-1}(x) = x - 4$. Notice that $f(5)=9$ and $f^{-1}(9)=5$.
• If a function's graph contains the point $(2, 8)$, the graph of its inverse must contain the point $(8, 2)$.
• To find the inverse of $y = 3x$, we swap $x$ and $y$ to get $x = 3y$, then solve for $y$ to find $y = \frac{x}{3}$.

Explanation

Think of an inverse function as a 'reverse' button. If one function takes you from A to B, its inverse takes you from B back to A. Their graphs are perfect reflections of each other across the line $y=x$.

Property

The log function $g(x) = \log_b x$ is the inverse of the exponential function $f(x) = b^x$. Because $f(x) = b^x$ and $g(x) = \log_b x$ are inverse functions for $b > 0$, $b \neq 1$,

Examples

• The statement $\log_3 81 = 4$ is equivalent to the exponential statement $3^4 = 81$.
• Using the inverse property, we can simplify $\log_8(8^5)$ directly to 5.
• Similarly, we can simplify $10^{\log_{10} 1000}$ directly to 1000.

Explanation

A logarithm answers the question: 'What exponent do I need to put on the base 'b' to get the number 'x'?' It is the opposite operation of raising a base to a power.

Property

A logarithmic equation is one where the variable appears inside of a logarithm. To solve:

1. Use the properties of logarithms to combine all logs into one log.
2. Isolate the log on one side of the equation.
3. Convert the equation to exponential form.
4. Solve for the variable.
5. Check for extraneous solutions.

Examples

• To solve $\log_5(x+3) = 2$, convert to exponential form: $5^2 = x+3$. This gives $25 = x+3$, so $x=22$.
• Solve $\log_2(x-1) + \log_2 3 = 4$. First, combine logs: $\log_2(3(x-1)) = 4$. Convert: $2^4 = 3x-3$. So $16 = 3x-3$, and $x=\frac{19}{3}$.
• To solve $\log_{10}(x) = -2$, convert to exponential form: $10^{-2} = x$. So $x = \frac{1}{100}$.

Explanation

To free a variable trapped inside a log, you must convert the equation to its exponential form. First combine logs, isolate the logarithm, then rewrite the equation to solve for the variable.

Property

The natural exponential function is $f(x) = e^x$ and the natural log function is $g(x) = \ln x = \log_e x$, where $e \approx 2.7182818245$.
Conversion Formula: $y = \ln x$ if and only if $e^y = x$.
Properties: For $x, y > 0$,

1. $\ln(xy) = \ln x + \ln y$
2. $\ln \frac{x}{y} = \ln x - \ln y$
3. $\ln x^k = k \ln x$

Also, $\ln e^x = x$ and $e^{\ln x} = x$.

Examples

• To solve the equation $e^x = 10$, take the natural log of both sides: $\ln(e^x) = \ln(10)$, which simplifies to $x = \ln(10)$.
• Simplify $\ln(e^2) + \ln(e^3)$. This is $2+3=5$. Alternatively, $\ln(e^2 \cdot e^3) = \ln(e^5) = 5$.
• To solve $\ln(x) = 4$, convert to exponential form: $x = e^4$.

Explanation

The natural log (ln) is a special logarithm with a base called 'e', an irrational number vital for describing continuous growth. It follows all the same rules as other logarithms.

Property

The function

describes exponential growth if $k > 0$, and exponential decay if $k < 0$. Here, $P_0$ is the initial amount, $t$ is time, and $k$ is the continuous growth or decay rate.

Examples

• A culture starts with 500 cells ($P_0=500$) and grows with $k=0.1$ per hour. After 3 hours, the population is $P(3) = 500e^{0.1 \times 3}$.
• A 100g sample of a substance decays with $k=-0.05$ per year. The amount remaining after 20 years is $P(20) = 100e^{-0.05 \times 20}$.
• If a town's population grew from 20,000 to 25,000 in 10 years, we find $k$ by solving $25000 = 20000e^{10k}$.

Explanation

This formula models things that grow or shrink at a rate proportional to their current size, like populations or radioactive substances. A positive 'k' means growth, while a negative 'k' means decay.