Exponential Equation
In short: An exponential equation is an equation with the unknown in an exponent, such as 7 · 3^x = 1,260, and it is solved by matching bases or by taking a logarithm of both sides.
Algebra 2AlgebraF-LE.A.4
What is Exponential Equation?
An exponential equation hides its unknown up in the exponent. Ordinary tools like dividing or taking roots cannot reach it there, so you need a different move: rewrite both sides with the same base, or use a logarithm to lower the exponent.
If both sides can be written as powers of one base, \(b^m = b^n\) forces \(m = n\). Otherwise isolate the power, then apply a log: \(b^x = c\) gives \(x = \frac{\ln c}{\ln b}\) for \(b > 0\), \(b \neq 1\) and \(c > 0\).
How to spot it
- An exponential equation has the variable in an exponent, as in 5^(2x – 1) = 125 or 2^x = 40.
- A growth or decay story asks when a quantity reaches a given amount.
- Both sides look like powers of 2, 3 or 10, which hints that the bases can be matched.
- Dividing by a coefficient leaves a single power equal to a plain number that is not an obvious power.
The method
- Isolate the exponential part of the exponential equation by dividing away any coefficient multiplying it.
- If both sides can be written with one base, set the exponents equal and solve.
- Otherwise take \(\ln\) of both sides and bring the exponent down as a multiplier.
- Divide to get the variable, then verify by substituting into the original equation.
Exponential Equation: solved example
Problem
Solve \(7\cdot 3^x = 1{,}260\). Give the answer to three decimal places.
Solution
Divide by 7 to isolate the power: \(3^x = 180\). Since 180 is not a power of 3, take the natural log: \(x\ln 3 = \ln 180\).
Then \(x = \frac{\ln 180}{\ln 3} \approx \frac{5.19296}{1.09861} \approx 4.727\). Check: \(3^4 = 81\) and \(3^5 = 243\), and 180 sits between them. The solution is \(x \approx 4.727\).
Try it
Question 1
Solve \(9^x = 27\).
Show the answer
Write both as powers of 3: \(3^{2x} = 3^3\), so \(2x = 3\) and \(x = \frac{3}{2}\).
Question 2
Solve \(2^x = 40\).
Show the answer
\(x = \frac{\ln 40}{\ln 2} \approx \frac{3.6889}{0.6931} \approx 5.322\).
Exponential Equation vs. Exponential Function
A function such as \(f(x) = 3^x\) is a rule that produces an output for each input, and its graph is a curve. The equation asks a question about it: for which input does the rule give a particular output? Solving means finding where the curve meets a horizontal line.
Keep going
Lessons
Worksheets and quizzes
- Exponential and Logarithmic Equations – Worksheet
- Exponential and Logarithmic Equations – Quiz
- Exponential Growth and Decay – Worksheet
- Exponential Growth and Decay – Quiz
Related words
- Logarithm
- Natural Logarithm (ln)
- Logarithm Rules
- Change of Base Formula
- Exponential Function
- Exponential Growth
Whole dictionary: A–Z · Algebra 2 · Algebra