Exponential and logarithmic equations and inequalities — AP Precalculus
1. Solving Exponential Equations ★★☆☆☆ ⏱ 4 min
An exponential equation has the general form $a^{f(x)} = b^{g(x)}$, where $a,b>0$ and $a,b \neq 1$. There are two core solution methods, depending on whether the bases can be rewritten to match.
If both sides can be rewritten with the same base, apply the one-to-one property to drop the base and solve the resulting polynomial equation directly. If bases cannot be matched, take the natural or common logarithm of both sides, apply the power rule $\ln a^b = b\ln a$ to bring the exponent down, then solve for $x$.
Exam tip: Always confirm what form the question asks for (exact vs approximate); AP Precalc almost always requires 3 decimal places for approximate answers, so double-check your rounding.
2. Solving Logarithmic Equations ★★★☆☆ ⏱ 4 min
To solve logarithmic equations, we use the one-to-one property for matching logarithms or rewrite the equation in exponential form using the definition of a logarithm. The most critical step is checking for extraneous solutions, which occur when a solution makes the argument of any original logarithm non-positive.
Exam tip: Always write domain restrictions from the original equation, not just the combined logarithm; it is possible for the combined argument to be positive even if an original argument is negative, leading to an invalid solution.
3. Solving Exponential and Logarithmic Inequalities ★★★☆☆ ⏱ 4 min
Exponential and logarithmic inequalities follow the same initial steps as equations: first find the domain, then rewrite to use the one-to-one property. The key difference is adjusting inequality direction based on whether the function is increasing or decreasing.
For base $b>1$, both $b^x$ and $\log_b x$ are strictly increasing, so inequality direction is preserved when dropping the base or logarithm. For $0<b<1$, both functions are strictly decreasing, so inequality direction is reversed when dropping the base or logarithm. The final solution set is the intersection of the simplified solution with the original domain.
Exam tip: If the question does not specify a form for the solution, use interval notation; it is universally accepted on the AP Precalc exam and less prone to notation errors.
4. AP-Style Practice Problems ★★★★☆ ⏱ 2 min
Common Pitfalls
Why: Students remember the one-to-one property but forget that only positive arguments are valid, so extraneous solutions are often left in.
Why: Students confuse the base rule for inequality direction, reversing the sign when it should be preserved for bases greater than 1.
Why: Dividing by a variable expression assumes it is non-zero, which eliminates any potential root at $x=0$.
Why: The power rule $\ln a^b = b \ln a$ only holds when $a>0$; squaring makes the argument positive even if $2x+1$ is negative, so solutions can be lost.
Why: The combined argument can be positive for values that make an original individual logarithm's argument negative, leading to an invalid solution.