2.13 Exponential and Logarithmic Equations and Inequalities

To solve exponential and logarithmic equations and inequalities, use exponent rules, log rules, and the fact that exponentials and logarithms undo each other. Always check your answers for extraneous solutions, since log arguments must stay positive.

Exponential and Logarithmic Equations Summary

In AP Precalculus Topic 2.13, exponential and logarithmic equations are usually about choosing the right rewrite. If the bases match, set the exponents equal. If the variable is in an exponent and the bases do not match, take $\ln$ or $\log$ of both sides and use the power rule. If the equation has logarithms, combine or rewrite them carefully while keeping the original domain restrictions.

The most important check is whether your answer is allowed in the original problem. Log arguments must be positive, contextual quantities may have limits, and inequalities can flip when you apply a decreasing exponential or logarithmic function with base between 0 and 1.

more resources to help you study

practice multiple choiceFRQ practice & scoringcheatsheetsscore calculator

Why This Matters for the AP Precalculus Exam

This topic pulls together everything you have learned about exponential and logarithmic functions in Unit 2 and turns it into a problem-solving skill. Exponential and logarithmic functions carry a large share of weight on the exam, so being able to solve these equations cleanly matters.

You will see these skills on both the multiple-choice and free-response sections. Some questions can be done by hand, while others let you use a graphing calculator to find points of intersection or numerical solutions. On free-response questions, answers without supporting work may not support a stronger score, so showing each step is important for clear exam work. Because exponential and logarithmic manipulation needs precision, skipping steps is where most errors sneak in.

Key Takeaways

• Use properties of exponents, properties of logarithms, and the inverse relationship between exponentials and logs to rewrite and solve equations and inequalities.
• A logarithm and an exponential with the same base undo each other: $\log_b c = a$ exactly when $b^a = c$.
• Always check for extraneous solutions, especially solutions that would make a log argument zero or negative.
• The identity $b^x = c^{(\log_c b)(x)}$ lets you rewrite an exponential expression with a different base when that is useful.
• To find the inverse of $f(x) = ab^{(x-h)} + k$ or $f(x) = a\log_b(x-h) + k$, reverse each operation in the opposite order it was applied.
• Logarithms have a restricted domain (argument greater than zero), so keep that in mind for both solving and inverses.

Solving Exponential and Logarithmic Equations and Inequalities

Properties of exponents, properties of logarithms, and the inverse relationship between exponential and logarithmic functions can all be used to solve equations and inequalities. These tools let you change the form of an equation into something easier to solve.

Using Exponent and Log Properties

Exponent rules like the product, quotient, and power properties help you combine or split exponential expressions. For example, if you have

$2^x \cdot 2^y = 2^z$

you can use the product property to write

$2^{x+y} = 2^z.$

Once the bases match, the exponents must be equal, so $x + y = z$.

Logarithm rules work the same way. The product rule lets you combine two logs with the same base:

$\log_2 x + \log_2 y = \log_2(xy).$

Combining logs into a single logarithm is often the first step in solving a logarithmic equation, because once you have a single log you can rewrite the equation in exponential form.

Using the Inverse Relationship

Because exponentials and logarithms are inverses, you can switch between the two forms to isolate a variable. For example, to solve

$2^x = 8$

rewrite it as a logarithm:

$x = \log_2 8 = 3.$

If the bases do not match nicely, take a logarithm of both sides instead. For an equation like $5^x = 17$, take $\log$ or $\ln$ of both sides and use the power rule to bring the exponent down.

Changing the Base

You can rewrite an exponential expression using a different base with the identity

$b^x = c^{(\log_c b)(x)}.$

This is handy when you want everything written with the same base, or with base $e$, so the rest of the problem is easier to manage.

Inequalities and Direction

Solving an inequality works much like solving the matching equation, but you have to watch the direction of the inequality. Exponential and logarithmic functions with a base greater than 1 are increasing, so applying them keeps the inequality direction the same. With a base between 0 and 1 the function is decreasing, which flips the inequality. Always think about whether the function you are applying is increasing or decreasing.

Extraneous Solutions

When you solve exponential and logarithmic equations, some answers you find may not actually be valid. These are called extraneous solutions, and they can show up from either algebraic steps or graphing.

The most common trap with logarithms is the domain. A logarithm is only defined when its argument is greater than zero, so any candidate solution that makes a log argument zero or negative must be thrown out.

For example, when solving

$\log(x+1) + \log(x-1) = \log 8$

you can combine the left side into $\log\big((x+1)(x-1)\big)$, set the arguments equal, and solve. After finding candidate values, plug each one back into the original equation. Any value that makes $x+1$ or $x-1$ zero or negative is not a real solution, since the log of that argument does not exist.

Always finish by checking your results against the domain and any limits from the context of the problem.

Building Inverses of Exponential and Logarithmic Functions

You can build the inverse of transformed exponential and logarithmic functions by reversing each operation. The idea is the same as undoing a chain of steps: reverse them in the opposite order they were applied.

Inverse of an Exponential Function

The function

$f(x) = ab^{(x-h)} + k$

is the general exponential function with shifts and a vertical stretch. To find its inverse, start from $y = ab^{(x-h)} + k$ and undo each piece:

1. Subtract $k$ from both sides: $\quad y - k = ab^{(x-h)}$

2. Divide both sides by $a$: $\quad \dfrac{y-k}{a} = b^{(x-h)}$

3. Take a logarithm (here $\ln$) of both sides: $\quad \ln\!\left(\dfrac{y-k}{a}\right) = (x-h)\ln b$

4. Solve for $x$: $\quad x = h + \dfrac{\ln\!\left(\dfrac{y-k}{a}\right)}{\ln b}$

This gives you the input $x$ that produces a given output $y$.

Inverse of a Logarithmic Function

The function

$f(x) = a\log_b(x-h) + k$

is the general logarithmic function with the same kinds of transformations. Starting from $y = a\log_b(x-h) + k$, undo each operation:

1. Subtract $k$ from both sides: $\quad y - k = a\log_b(x-h)$

2. Divide both sides by $a$: $\quad \dfrac{y-k}{a} = \log_b(x-h)$

3. Rewrite in exponential form (raise $b$ to both sides): $\quad b^{(y-k)/a} = x - h$

4. Add $h$ to both sides: $\quad x = b^{(y-k)/a} + h$

So the inverse is $f^{-1}(x) = b^{(x-k)/a} + h$ once you swap the roles of $x$ and $y$. Notice that the exponential inverse uses a log step, and the logarithmic inverse uses an exponential step, which matches the inverse relationship between the two function types.

How to Use This on the AP Precalculus Exam

Problem Solving

• Decide first whether to match bases or take a logarithm of both sides. Matching bases is cleaner when both sides can be written with the same base.
• When a variable is stuck in an exponent, take $\ln$ or $\log$ of both sides and use the power rule to bring it down.
• Combine logs into a single logarithm before rewriting a logarithmic equation in exponential form.
• For inequalities, check whether the base is greater than 1 (direction stays) or between 0 and 1 (direction flips).

Free Response

• Show each step of your manipulation. Answers without supporting work may not support a stronger score, so write out your log and exponent moves clearly.
• State any solutions you reject and why, such as a value that makes a log argument negative.
• If a question allows a calculator, you can confirm a solution by finding the intersection of the two sides graphed as separate functions.

Common Trap

• Forgetting to check the domain after solving a log equation. Always test candidate solutions in the original equation.

Common Misconceptions

• "Every algebraic solution is valid." Some answers are extraneous. A value that makes a log argument zero or negative is not a real solution, even if the algebra looked fine.
• "Inequalities always keep the same direction." With an exponential or logarithmic base between 0 and 1, the function is decreasing, so the inequality direction flips.
• "$\log(x+1) + \log(x-1)$ is the same as $\log(x+1)(x-1)$ with no restrictions." The combined form is only valid where both original arguments are positive, so you still have to respect the original domain.
• "You can only solve $b^x = c$ when the bases match." If they do not match, take a logarithm of both sides and use the power rule to bring the exponent down.
• "Finding an inverse just means flipping a sign." Building an inverse means reversing every operation in the opposite order, including the log or exponential step that undoes the function type.

Related AP Precalculus Guides

• 2.8 Inverse Functions
• 2.10 Inverses of Exponential Functions
• 2.4 Exponential Function Manipulation
• 2.7 Composition of Functions
• 2.11 Logarithmic Functions
• 2.3 Exponential Functions