Exponential Equations

To solve an exponential equation, isolate the exponential expression and choose a method that respects its structure.

Recognizing the equation type

An exponential equation contains a variable in an exponent, such as

This differs from $x^3=81$, where the variable is the base. Ordinary polynomial techniques do not move a variable down from an exponent; use equivalent bases, graphs, or logarithms.

Method 1: write a common base

If both sides can be expressed with the same positive base other than $1$, equate the exponents because the exponential function is one-to-one.

For example,

so

This method gives exact answers quickly when powers are recognizable.

Worked example with exponent laws

Substitution in the original equation confirms the equality.

Isolate before solving

Consider

First undo the outside operations:

Then write $8=2^3$, giving $x=3$. Trying to apply exponent rules before isolating $2^x$ often creates invalid steps.

Method 2: graph the intersection

An equation $f(x)=g(x)$ can be solved by graphing $y=f(x)$ and $y=g(x)$ and finding their intersection coordinates. This is useful when no easy common base exists or when a course expects technology-assisted solutions.

Graphing provides an estimate, so set a sensible viewing window and verify the reported $x$-value by substitution. Multiple intersections are possible in equations that combine exponential and other functions.

Method 3: use logarithms

Logarithms undo exponentiation. For

take a logarithm of both sides:

Bring down the exponent:

Natural logarithms work equally well: $x=\ln7/\ln2$.

Worked application: reaching a target

An account follows $A(t)=2000(1.05)^t$. To find when it reaches $3000$:

Then

The continuous model reaches the target after about $8.31$ years. If interest is credited only at year-end, the first whole-year balance at or above the target must be checked separately.

Extraneous and impossible cases

For real $x$, a positive exponential expression $b^x$ is always positive when $b>0$. Therefore $2^x=-5$ has no real solution. Similarly, after isolating an exponential expression, a nonpositive right side signals no real solution.

Transformations can change the range, so reason using the full expression rather than assuming every exponential equation has a solution.

Checking and communicating precision

Substitute an exact or unrounded value into the original equation. When using logarithms, keep the calculator value until the final step. State whether the result represents an instant, a completed period, or a rounded count.

A graph or table can provide a separate reasonableness check.

Common mistakes

Equating exponents when bases differ. First rewrite with a genuinely common base.

Distributing an exponent over addition. $(a+b)^x$ is not $a^x+b^x$.

Taking a logarithm of only one term. Apply an operation to both complete sides of the equation.

Rounding too early. It can shift a predicted time or threshold.

Ignoring the range. A positive exponential cannot equal a negative target before a vertical shift is considered.

Quick self-check

• Is the exponential expression isolated?
• Can both sides be rewritten with the same base?
• If not, is graphing or a logarithm appropriate?
• Did I use exponent and log laws legally?
• Does the solution verify in the original equation?
• Does its rounding fit the context?

Related topics

• Exponential Functions
• Exponents
• Logarithmic Functions
• Compound Interest