Exponential Equation

Sommaire

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.

3^x climbs to 180 at x = 4.727, the solution of 3^x = 180.

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.

Exponential Function

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