Exponential equations

Exponential equations

Choose and justify a method for solving exponential equations, from matching bases to taking logarithms and substituting a new variable. This precalculus practice emphasizes complete real solution sets rather than isolated guesses.

Work with reciprocal bases, shifted exponentials, different bases on opposite sides and sums sharing a common exponential factor. Quadratic-in-exponential equations require a positive substituted variable: a polynomial root can be negative or zero without corresponding to any real exponent. Other tasks involve a quadratic exponent, a restricted rational exponent, a continuous model and a parameter that changes the number of solutions.

Prerequisites are exponent rules, logarithm laws, change of base and quadratic equations. Single-select and multiselect questions develop from recognizing possible outputs to solving and checking several branches. Exact logarithmic answers are retained; a scientific calculator is allowed only for the explicitly requested approximation.

Write the domain and isolate or factor the exponential before applying logarithms. Detailed explanations verify candidates in the original equation, show why substitutions must stay positive and distinguish a quadratic in the exponent from a quadratic in the exponential expression. Questions are independent, so solving one is never a prerequisite for answering another.