Solving Exponential Equations

Every question of the form “how long until…?” in an exponential model — how long until an investment doubles, until a population reaches a million, until a drug falls below a safe dose — ends in an exponential equation, with the unknown sitting in the exponent. Solving exponential equations is the payoff for everything earlier in this chapter: the definition of a logarithm, the power rule and the change of base formula all come together here.

Two Methods for Solving Exponential Equations

Try the common-base method first. It gives exact answers with no calculator. When it fails — as it does for , since is not a power of — switch to logarithms.

Method 1: Rewrite With a Common Base

Because exponential functions are one-to-one, equal powers of the same base must have equal exponents:

The last row shows that equating exponents can produce a quadratic. Solve it as usual — both answers are valid, since every real number is an allowed exponent.

Method 2: Take the Logarithm of Both Sides

When there is no common base, use this sequence:

1. Isolate the exponential expression on one side.
2. Take (or ) of both sides.
3. Use the power rule to bring the exponent down.
4. Solve the resulting linear equation, then evaluate with a calculator.

For :

Taking the log of both sides is legitimate because both sides are equal positive numbers, and equal numbers have equal logarithms. If the base is , use ; if it is , use — each cancels its own base in one step.

Example 1: Solving Exponential Equations Using Logarithms

The last row is the classic doubling-time question: money growing at a year doubles in a little over years, whatever the starting amount.

Exponential Equations With Different Bases on Both Sides

When the unknown appears in exponents on both sides with bases that do not match, take logs of both sides and collect the terms:

The step people miss is the distribution in the second line: must become before the terms can be gathered.

Exponential Equations in Quadratic Form

An equation such as is a quadratic in disguise, since . Substitute , as in equations reducible to quadratic form:

So or , giving or . If either factor had given equal to a negative number, that branch would have no solution and would be discarded.

Worked Example

Solve , giving the answer to four decimal places.

Step 1 — isolate the exponential term by subtracting :

Step 2 — divide by :

Step 3 — take the natural log of both sides; cancels :

Step 4 — divide by :

Step 5 — check by substituting:

So .

Common Mistakes to Avoid

• Taking logs before isolating. cannot be simplified — there is no rule for the log of a sum. Get the exponential alone first.
• Combining the coefficient with the base. is not . Divide by first.
• Simplifying to . They are different numbers: versus . Divide the two logs as numbers.
• Accepting a negative number. Exponentials are always positive, so that equation has no solution.

Practice Problems

Work each one before opening the answer.

Problem 1. Solve .

Problem 2. Solve to four decimal places.

Problem 3. Solve to four decimal places.

Problem 4. Solve to four decimal places.

Problem 5. Solve .

Problem 6. An account grows at per year. How many years does it take to double?

The mirror-image problem has the unknown inside a logarithm rather than in an exponent. The solving logarithmic equations lesson handles those — and adds the one extra step they need, checking for extraneous solutions.