2.13 Exponential and Logarithmic Equations and Inequalities

Introduction: Finding the "When" and "How Much"

Welcome to one of the most practical chapters in AP Precalculus! Up to this point, you’ve learned what exponential and logarithmic functions look like and how to move their parts around using properties. Now, we are putting those skills to work. Solving exponential and logarithmic equations and inequalities is how we answer real questions, like "How long will it take for this population to triple?" or "At what intensity does a sound become painful to the human ear?"

Whether you are solving these by hand (without a calculator) or using technology to find an intersection point, the goal is the same: finding the specific value of \( x \) that makes the statement true.

Part 1: Solving Exponential Equations

An exponential equation is one where the variable is sitting up in the exponent, like \( 2^x = 10 \). There are two main ways to tackle these depending on whether you have a calculator or not.

Strategy A: The "Common Base" Method (No Calculator)

If you can rewrite both sides of the equation so they have the same base, the exponents must be equal. This is based on the one-to-one property of exponential functions.

The Rule: If \( b^x = b^y \), then \( x = y \).

Example: Solve \( 9^{x+1} = 27^x \).
1. Recognize that both 9 and 27 are powers of 3: \( (3^2)^{x+1} = (3^3)^x \).
2. Simplify exponents: \( 3^{2x+2} = 3^{3x} \).
3. Set the exponents equal: \( 2x + 2 = 3x \).
4. Solve for \( x \): \( x = 2 \).

Strategy B: Taking the Logarithm of Both Sides

When the bases can’t be easily matched (like \( 5^x = 12 \)), we use logarithms to "bring the variable down." Usually, we use the natural log (\( \ln \)) or common log (\( \log \)).

Step-by-Step:
1. Isolate the exponential expression.
2. Take the log of both sides: \( \ln(5^x) = \ln(12) \).
3. Use the Power Property: \( x \cdot \ln(5) = \ln(12) \).
4. Divide to solve: \( x = \frac{\ln(12)}{\ln(5)} \).

Quick Review: Remember from Topic 2.12 that \( \frac{\ln(12)}{\ln(5)} \) is the same as \( \log_5(12) \). On the AP Exam, you might see the answer in either form!

Key Takeaway: Use common bases if you see them; otherwise, logs are your best friend for getting variables out of exponents.

Part 2: Solving Logarithmic Equations

Logarithmic equations have the variable inside the log argument, like \( \log_3(x) = 4 \). Just like exponential equations, we have two primary paths.

Strategy A: The "Log = Log" Method

If you have a single logarithm of the same base on both sides, their arguments must be equal.
The Rule: If \( \log_b(M) = \log_b(N) \), then \( M = N \).

Strategy B: Converting to Exponential Form

If you have a log equal to a constant, rewrite it as an exponential.
The Rule: If \( \log_b(x) = c \), then \( b^c = x \).

The "Golden Rule" of Logs: Check for Extraneous Solutions!

Important: The domain of a logarithmic function is \( (0, \infty) \). You cannot take the log of a negative number or zero. Always plug your answer back into the original equation to ensure the argument is positive. If it makes the inside of a log negative, you must discard that solution!

Example: \( \log_2(x) + \log_2(x-2) = 3 \).
1. Combine logs: \( \log_2(x(x-2)) = 3 \).
2. Rewrite as exponential: \( 2^3 = x^2 - 2x \).
3. Solve the quadratic: \( 8 = x^2 - 2x \implies x^2 - 2x - 8 = 0 \implies (x-4)(x+2) = 0 \).
4. Potential answers: \( x = 4 \) and \( x = -2 \).
5. Check: Plugging in \( 4 \) works (\( \log_2(4) \) is fine). Plugging in \( -2 \) fails (\( \log_2(-2) \) is undefined). Final answer: \( x = 4 \).

Key Takeaway: Always check your answers! Logarithms are "picky eaters" and only accept positive values.

Part 3: Solving Inequalities

Solving inequalities like \( e^x > 5 \) or \( \ln(x) \leq 2 \) follows the same logic as equations, but with a focus on intervals.

Analytical Approach (By Hand)

1. Solve the equation as if it were an equals sign to find the critical value (the boundary).
2. Consider the domain of the function (especially for logs!).
3. Test values on a number line or use your knowledge of whether the function is increasing or decreasing.

Technological Approach (Graphing Calculator)

On the calculator-active portion of the AP Exam (Section I Part B and Section II Part A), you can solve inequalities graphically:
1. Graph the left side as \( y_1 \) and the right side as \( y_2 \).
2. Find the intersection point using the calculator's "Intersect" tool.
3. Look at the graph to see where the curve is above or below the line.
4. Pro Tip: If you are looking for where \( f(x) > g(x) \), you are looking for the \( x \)-intervals where the graph of \( f \) is visually higher than the graph of \( g \).

Key Takeaway: Inequalities are about intervals. Don't forget to check if the function's domain restricts your interval (e.g., \( x \) must be greater than 0 for logs).

Part 4: Calculator Skills for the AP Exam

The AP Precalculus exam assumes you are proficient with a graphing calculator for certain tasks. For this chapter, you should be able to:

• Find Zeros: If you rewrite an equation as \( f(x) = 0 \), use the "Zero" or "Root" tool to find the solution.
• Find Intersections: Graph two functions and find where they cross. This is often faster than solving for zero.
• Numerical Solutions: Use the "Solver" feature to find a specific value of \( x \).
• Maintain Accuracy: Never round your numbers in the middle of a problem. Store the full decimal in your calculator and only round your final answer to three decimal places (the AP standard).

Did you know? On the AP Exam, some questions require a calculator because the algebra is impossible to do by hand, while others forbid it to test your symbolic fluency. Practice both ways!

Common Mistakes to Avoid

1. Distributing Logs: \( \log(A + B) \) is NOT \( \log(A) + \log(B) \). There is no rule for the log of a sum.
2. Base Confusion: Using the wrong base when converting to exponential form (e.g., thinking \( \ln(x) \) has a base of 10 instead of \( e \)).
3. Forgetting Domain: Not checking for extraneous solutions in logarithmic equations.
4. Rounding Early: Rounding \( \ln(2) \) to \( 0.69 \) too early in a problem can lead to a wrong final answer. Keep those decimals until the end!

Summary & Key Takeaways

• Exponential Equations: Try to match bases first. If you can't, take the log of both sides and use the Power Property to pull the variable down.
• Logarithmic Equations: Use properties to condense into a single log, then rewrite in exponential form (\( b^c = x \)).
• The Domain Check: Always verify that your solutions don't result in taking the log of a non-positive number.
• Inequalities: Use critical values and domain boundaries to define intervals. Graphs are incredibly helpful for visualizing these.
• Calculator Standard: Round final answers to three decimal places.