Exponential Equation – Solving Steps, Examples (Algebra 2)

Contents

Key takeaway: An exponential equation has the unknown in an exponent, and it is solved either by matching bases on both sides or by taking a logarithm of both sides.

Grade 11 – Algebra 2F-LE.A.4Expressions and Equations

Definition of Exponential Equation

Definition

An exponential equation is an equation where the variable sits in the exponent, such as \(7^{x-2}=343\) or \(4\cdot 3^x=500\). Ordinary algebra cannot reach an exponent directly, so you need one of two tools.

When both sides can be written with the same base, the exponents must be equal, and you solve a simple equation. When they cannot, you isolate the power and apply a logarithm, which brings the exponent down to ground level. Either way, you first get the exponential piece alone before doing anything else.

Formula and notation

Two core moves, for \(b>0\), \(b\neq 1\):

\[b^{m}=b^{n}\implies m=n, \qquad b^{x}=k\implies x=\frac{\ln k}{\ln b}.\]

The second needs \(k>0\); an equation such as \(b^x=-5\) has no real solution.

The solution is the x-coordinate where the curve reaches the height 500.

Key facts

In an exponential equation, isolate the power first: divide by any coefficient before using a logarithm.
Matching bases is faster when numbers are clean powers, like 343 = 7³.
A power with a positive base is always positive, so it can never equal a negative number.
Substitute the answer back to confirm it, especially after rounding.

Exponential Equation example

Problem

Solve \(4\cdot 3^x=500\) and give the answer to two decimal places.

Isolate the power by dividing both sides by 4: \(3^x=125\).
Take the natural logarithm of both sides: \(x\ln 3=\ln 125\).
Divide: \(x=\frac{\ln 125}{\ln 3}\approx\frac{4.8283}{1.0986}\approx 4.39\).
Check: \(3^{4.39}\approx 124.5\) and \(4\times 124.5\approx 498\), close to 500 given the rounding.

The solution is \(x\approx 4.39\).

Common mistake

Watch out

Selin rewrites \(4\cdot 3^x=500\) as \(12^x=500\), multiplying the coefficient into the base.

Why it is wrong

The 4 multiplies the whole power; it is not part of the base. At \(x=2\) the left side is \(4\cdot 9=36\), but \(12^2=144\). In any exponential equation, divide by the coefficient first, then deal with the base.

Related terms and lessons

Algebra 2 · Solving Exponential and Logarithmic EquationsLesson Worksheet Quiz

Browse the glossary: A–Z list · Algebra 2 vocabulary · Expressions and Equations

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