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All courses Algebra II · F-LE.4 36 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 \(3\cdot~2^{4t}=48\)?
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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 power

\[3*2^{4t}=48~->~2^{4t}=16\]

Divide both sides by the nonzero coefficient 3 before working with the exponent.

02

Write both sides with base two

\[16=2^4~->~2^{4t}=2^4\]

Equal positive powers with the same valid base have equal exponents, so 4t=4.

03

Solve and check

\[4t=4~->~t~=~1;~\text{check}:~3*2^{4*1}=3*16=48\]

The exact substitution reproduces the original right side, confirming the solution.

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Another way

  1. Take logarithms after isolating the power: 4t*ln(2)=ln(16)=4ln(2), giving t=1.

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Common mistake

Do not divide the exponent by 3. The 3 is an outside multiplier, so divide the equation by 3 first; only then compare exponents.

Solution walkthrough video