Topic 2.13 Exponential and Logarithmic Equations and Inequalities: solve exponential and logarithmic equations and inequalities using inverse operations, the logarithm properties, and checks for extraneous solutions.

What this topic is asking

The College Board (Topic 2.13) wants you to solve exponential and logarithmic equations and inequalities. The core technique is using inverse operations: take a logarithm to bring a variable down from an exponent, or exponentiate to free a variable from inside a logarithm. You must use the logarithm properties to combine terms, and you must check for extraneous solutions because logarithms have restricted domains.

Solving exponential equations

For example, $5^x = 30$ gives $x\ln 5 = \ln 30$, so $x = \frac{\ln 30}{\ln 5}$.

Solving logarithmic equations

The condense-then-exponentiate routine converts a logarithmic equation into an ordinary algebraic one, but the domain restriction is what forces the final check.

Checking for extraneous solutions

Inequalities

Solving an exponential or logarithmic inequality follows the same steps, but you track the direction of the inequality. For a base greater than $1$, the exponential and logarithmic functions are increasing, so applying the log (or exponentiating) preserves the inequality direction. You also intersect the algebraic solution with the domain restrictions: for a logarithmic inequality, the arguments must stay positive, which can trim the solution set.

A point that separates careful work is that the domain check is not optional housekeeping; it is part of solving. The algebra of condensing and exponentiating can introduce values that were never legal inputs, so the original domain (every logarithm argument positive) defines which candidates can possibly count. Building the domain restriction in from the start, by writing down $x > 0$ and $x - 2 > 0$ before solving, often reveals immediately that a candidate like $x = -2$ cannot survive, which is the reasoning the free-response rubric rewards.

A second clarifying idea is that taking a logarithm and exponentiating are the inverse moves that unlock each equation type. An exponential equation hides the variable in an exponent, so a logarithm (the inverse) frees it; a logarithmic equation hides the variable inside a log, so exponentiating (the inverse) frees it. Recognizing which inverse operation matches the equation, rather than guessing, makes the solving step automatic and connects directly to the inverse relationship from Topic 2.10.

Try this

Q1. Solve $2^{x} = 16$. [1 point]

• Cue. $16 = 2^4$, so $x = 4$ (matching bases avoids logs here).

Q2. Solve $\log(x) = 3$. [1 point]

• Cue. Rewrite as $x = 10^3 = 1000$.