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All courses Math III · F-LE.4 29 of 55
Use logarithms to solve exponential equations of the form ab^(ct)=d and evaluate with technology.

Solve an exponential equation of the form a(b)^(ct)=d by isolating the exponential factor and taking logarithms

Problem
What is the solution to the exponential equation 3*2⁴t=48?
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Hint

Start by isolating the exponential expression. Divide both sides by \(3\) before working with the exponent.

After dividing by \(3\), the equation is \(2^(4t)=16\). Rewrite \(16\) as \(2^4\), then compare the exponents because the bases are the same.

Solution walkthrough

01

Isolate the exponential factor

\[\begin{aligned} 3*2^(4t)=48 \\ 2^(4t)=16 \end{aligned}\]

The coefficient \(3\) is outside the exponential expression, so dividing both sides by \(3\) leaves the power of \(2\) by itself.

02

Rewrite with the same base

\[\begin{aligned} 16=2^4 \\ 2^(4t)=2^4 \end{aligned}\]

The right side is a power of the same base as the left side, so the equation can be written with base \(2\) on both sides.

03

Set the exponents equal

\[4t=4\]

Equal powers with the same positive base have equal exponents, so \(4t\) must equal \(4\).

04

Solve for `t`

\[t=1\]

Divide both sides by \(4\) to isolate \(t\).

05

Check the solution

\[3*2^(4(1))=3*2^4=3*16=48\]

Substitution confirms the solution because the original left side becomes \(48\).

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Another way

  1. Use logarithms after isolation: \(4t\ln(2)=\ln(16)\), which also gives \(t=1\).

!

Common mistake

A common mistake is to compare exponents before dividing by \(3\). The equation must first be rewritten as \(2^(4t)=16\), not \(2^(4t)=48\).