Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 2 exponential and logarithmic functions? Then this Algebra 2 lesson on Using Logs to Solve Exponent Equations explains each idea step by step, with worked examples.
Quick answer: To solve an exponential equation, isolate the power and take a logarithm of both sides so the exponent comes down: if 5 · 3^x = 45, then 3^x = 9 and x = ln 9 ÷ ln 3 = 2. To solve a logarithmic equation, combine the logs, rewrite in exponential form, and discard answers that make an argument negative.
Standards: F-LE.A.4, A-SSE.B.3c · Level: Grade 11 Algebra 2
How do you solve an equation with the unknown in the exponent?
Isolate the power, then apply a logarithm to both sides. The power rule brings the exponent down so that it can be solved like a linear equation.
- Isolate the exponential expression: a · bct = d becomes bct = d/a.
- Take log or ln of both sides: ct · ln b = ln(d/a).
- Divide: t = ln(d/a) ÷ (c ln b).
- Check in the original equation.
Solve 4 · 2x = 80. Divide by 4: 2x = 20. Then x ln 2 = ln 20, so x = ln 20 ÷ ln 2 ≈ 2.9957 ÷ 0.6931 ≈ 4.32.
How do you solve a logarithmic equation?
Combine the logs into one, rewrite it in exponential form, and solve. A logarithm is only defined for positive arguments, so every answer must be tested in the original equation.
- logb M = k ⟺ M = bk
- logb M = logb N ⟹ M = N
Solve log3 x + log3(x − 8) = 2. Combine: log3(x² − 8x) = 2, so x² − 8x = 9. Then (x − 9)(x + 1) = 0. The value x = −1 makes the argument negative, so x = 9.
Writing ln(a + b) as ln a + ln b, or taking the log of only one term. Take the logarithm of an entire side, after isolating the power.
What models use these equations?
Compound interest A = P(1 + r/n)nt, continuous growth A = Pert and half-life N = N0(1/2)t/h all have the time t in an exponent. Logarithms solve for t.
Use ln when the base is e, and either log or ln for other bases. Keep full calculator values until the last step, then round once.
- Isolate the exponential expression before taking logarithms.
- The power rule turns bct = d into ct · ln b = ln d.
- logb M = k is equivalent to M = bk.
- Always check logarithmic solutions: arguments must be positive.
- Time in growth, decay, interest, pH and decibel problems is found with logarithms.
Using Logs to Solve Exponent Equations: frequently asked questions
When should I use ln instead of log?
Either works for any exponential equation because the ratio of the two logs is the same. Use ln when the base is e, and use log when the base is 10. Mixing them in one fraction is wrong.
What is an extraneous solution in a log equation?
It is a value that satisfies the simplified algebra but makes an argument of a logarithm zero or negative in the original equation. Such a value must be rejected after checking.
Why do I need to isolate the power first?
The logarithm rule applies to a single power. In 4 · 2x = 80 the factor 4 sits outside, so taking ln of both sides without dividing first gives a wrong equation.
How do I find doubling time?
Set the final amount equal to twice the starting amount and solve for time. For continuous growth at rate r the doubling time is ln 2 ÷ r, and the starting amount cancels.
Can exponential equations be solved without logarithms?
Yes, when both sides can be written with the same base, such as 4x+1 = 64, which gives x + 1 = 3. Otherwise logarithms, a graph or a table are required.