Solving exponential equations (incl. with logs)
Solving Exponential Equations
Solving Exponential Equations is a topic in Exponential & Logarithmic Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.5(D), which requires students to solve exponential equations of the form y = ab to the x.
Exponential equations are solved by matching bases and equating exponents, or by taking a logarithm of both sides.
Theory
To solve an exponential equation, get the variable out of the exponent:
- Same base: rewrite both sides with equal bases, then equate exponents.
- Different bases: take a logarithm of both sides and use the power law.
Log both sides:
How to solve
- Isolate the exponential term.
- If bases can match, equate exponents.
- Otherwise take \(\log\) (or \(\ln\)) of both sides.
- Use the power law and solve for \(x\).
Write \(16=2^4\) and equate exponents.
| \(2^x\) | \(=\) | \(2^4\) |
| \(x\) | \(=\) | \(4\) |
Take a log and use the power law.
| \(x\log 3\) | \(=\) | \(\log 20\) |
| \(x\) | \(=\) | \(\dfrac{\log 20}{\log 3}\approx2.73\) |
\(125=5^3\).
| \(x+1\) | \(=\) | \(3\) |
| \(x\) | \(=\) | \(2\) |
Take the natural log.
| \(2x\) | \(=\) | \(\ln 10\) |
| \(x\) | \(=\) | \(\dfrac{\ln 10}{2}\approx1.15\) |
Common pitfalls
Frequently asked questions
How do you solve \(2^x=8\)?
Write \(8=2^3\) and equate exponents: \(x=3\).
What if the bases don't match?
Take a logarithm of both sides and use the power law.
Which base of log should you use?
Any — base 10 or \(\ln\); use \(\ln\) for base \(e\).
Do you isolate the power first?
Yes — separate the exponential term before taking a log.