Pre-Calculus Exponential and logarithmic functions (advanced)

Solving advanced exponential equations

Q: Which logarithm should you use?

Any base works; the natural log is convenient because it is on every calculator.

Q: Why take the log of both sides?

The power rule brings the variable down out of the exponent so you can solve for it.

20 practice questions 0 video lessons Theory + worked examples

Solving Exponential Equations

California Pre-Calculus • Standard F-LE.4 • Exponential & Logarithmic Functions

Solving Exponential Equations is a topic in Exponential & Logarithmic Functions in the California Common Core State Standards. It is aligned to Standard F-LE.4, which requires students to solve exponential equations, expressing the solution using logarithms.

Solving an exponential equation means matching bases to equate exponents, or taking a logarithm of both sides to bring the variable down.

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Practice questions

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Theory

An exponential equation has the unknown in an exponent. Two strategies cover almost every case:

Same base: rewrite both sides with a common base, then set the exponents equal.
Different bases: take a logarithm of both sides and use the power rule \(\log b^x=x\log b\) to bring the variable down.

Isolate the exponential first. Get \(b^{\,\text{stuff}}\) alone on one side before matching bases or taking a log.

The two strategies for an exponential equation.

Graphically, \(2^x=5\) is where the curve meets \(y=5\).

The core moves:

\[b^{\,u}=b^{\,v}\Rightarrow u=v,\qquad b^x=c\Rightarrow x=\log_b c=\dfrac{\ln c}{\ln b}\]

equal bases give equal exponents; otherwise x equals log base b of c

Any log works — use \(\ln\) so the answer is ready for a calculator.

How to solve an exponential equation

Isolate the exponential expression.
Same base? Rewrite and equate exponents.
Otherwise take \(\ln\) of both sides.
Bring the exponent down with the power rule and solve.

Example 1 — Same base

Solve \(2^x=32\).

Solution

Write both sides with base 2, then equate exponents.

\(2^x\)	\(=\)	\(2^5\)
\(x\)	\(=\)	\(5\)

x equals 5

Example 2 — Take a log

Solve \(3^x=20\).

Solution

The bases can't be matched, so take \(\ln\) of both sides and use the power rule.

\(\ln 3^x\)	\(=\)	\(\ln 20\)
\(x\ln 3\)	\(=\)	\(\ln 20\)
\(x\)	\(=\)	\(\dfrac{\ln 20}{\ln 3}\approx 2.73\)

x is about 2.73

Example 3 — Base e

Solve \(e^{2x}=7\).

Solution

Take the natural log, which undoes \(e\).

\(2x\)	\(=\)	\(\ln 7\)
\(x\)	\(=\)	\(\dfrac{\ln 7}{2}\approx 0.973\)

x is about 0.973

Example 4 — Isolate first

Solve \(5\cdot 2^x=80\).

Solution

Divide by 5 to isolate the power, then match bases.

\(2^x\)	\(=\)	\(16\)
\(2^x\)	\(=\)	\(2^4\)
\(x\)	\(=\)	\(4\)

x equals 4

Common pitfalls

Isolate before taking a log. In \(5\cdot 2^x=80\), divide by 5 first.

Use the power rule after logging. \(\ln b^x=x\ln b\) is what frees the variable.

Keep exact or round sensibly. \(\dfrac{\ln 20}{\ln 3}\) is exact; a decimal is an approximation.

Frequently asked questions

How do you solve an exponential equation?

Isolate the exponential, then either match bases and equate exponents, or take a logarithm of both sides and use the power rule.

When can you just equate exponents?

When both sides can be written with the same base. Then equal bases force equal exponents.

Which logarithm should you use?

Any base works; the natural log \(\ln\) is convenient because it is on every calculator.

Why take the log of both sides?

The power rule \(\ln b^x=x\ln b\) moves the variable out of the exponent so you can solve for it.

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Solving advanced logarithmic equations

More in Exponential and logarithmic functions (advanced)

Solving advanced exponential equations

Solving Exponential Equations

📖 Theory

How to solve an exponential equation

Common pitfalls

Frequently asked questions

Theory