Solving Exponential Equations
Take logs of both sides to bring the exponent down.
About this topic
To solve an exponential equation, try to write both sides with the same base and equate the exponents, as in 2ˣ = 32 giving x = 5. When that is not possible, take the logarithm of both sides and use the power rule to bring the exponent down.
For 3ˣ = 20, x = log 20 / log 3, about 2.73. For 5 · 2ˣ = 80, first divide by 5 to isolate the exponential, giving 2ˣ = 16 and x = 4. When the unknown is in the exponent with a sum, like 1000(1.05)ᵗ = 2000, isolate the power to get 1.05ᵗ = 2, and then t = log 2 / log 1.05, about 14.2 years.
This is how you find doubling time, loan payoff time, the age of a fossil, or when a medicine falls below a safe level.
Always isolate the exponential before taking logs. A mistake is to take the log of each term, as in splitting a sum. Another is to forget that log of both sides must use the same base. Check by plugging the answer back in, and round only at the end to keep accuracy.
Key ideas
- Isolate the exponential term
- Apply a log
- Divide to solve
Quick check
Use cases: where Solving Exponential Equations helps
- Homework: review the key idea behind Solving Exponential Equations before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Solving Exponential Equations without notes.
- Visual learning: play with the interactive lab to see how Solving Exponential Equations changes when you adjust its inputs.
- Teaching a friend: explain Solving Exponential Equations in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Solving Exponential Equations?
Take logs of both sides to bring the exponent down.
What will I learn in Solving Exponential Equations?
Isolate the exponential term. Apply a log. Divide to solve.
Is Solving Exponential Equations part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Solving Exponential Equations lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Solving Exponential Equations?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.