Algebra 2 · Unit 7 · Topic 7.5
Solving Exponential Equations and Interpreting Time
Taking logarithms solves equations where the unknown is an exponent, provided the quantities being logged are positive.
Definition and explanation
Taking logarithms solves equations where the unknown is an exponent, provided the quantities being logged are positive. Express units for the resulting time and distinguish an exact solution from its rounded approximation.
Worked example
Solve 200(1.08)ᵗ = 300. Divide by 200, then take logs to obtain t = ln(1.5)/ln(1.08), approximately 5.27 periods. If balances are only reported after whole annual periods, the first reported year at or above 300 is year 6.
Common mistake
A continuous-time solution does not automatically identify the first whole reporting period.
Key takeaway
Taking logarithms solves equations where the unknown is an exponent, provided the quantities being logged are positive.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.