Solve problems involving exponential equations

Alberta Grade 12 Mathematics

Relations and Functions

Use equal bases, graphs, and context to find an unknown exponent.

An exponential equation has a variable in an exponent. For example, in

2^x=16

, the unknown value is the exponent

x

. Since

2^4=16

, the solution is

x=4

. You already know how to evaluate powers and solve simple equations. This lesson builds on those skills. First look for a way to write both sides with the same base. If that is not practical, use a graph or table to estimate a solution. In the equal-base method, the common base must be positive and cannot equal

1

. Always check that a solution fits both the equation and its context.

What you will learn

Recognize an exponential equation and identify its unknown exponent.
Solve equations exactly by rewriting powers with the same base.
Use a graph or table to estimate a solution when equal bases are not practical.
Check answers in the original equation and interpret them in context.

1. Review powers and identify the unknown

A power has a base and an exponent. In

3^4

, the base is

3

and the exponent is

4

. The expression means

3\times3\times3\times3

, which equals

81

. An exponential equation is an equation in which a variable appears in an exponent, such as

3^x=81

.

Two useful power rules are

a^0=1

for

a\ne0

, and

(a^m)^n=a^{mn}

. The second rule says to multiply the exponents when a power is raised to another power. For example,

(2^2)^x=2^{2x}

. These rules can help rewrite an equation before solving it.

For the equal-base method, use a common base that is positive and not equal to

1

. With such a base, different exponents give different powers. Therefore, if the powers are equal, their exponents must be equal.

a^m=a^n\Rightarrow m=n\quad(a>0,\ a\ne1)

The base is the number being raised to a power; the exponent tells the power.
An exponential equation has a variable in an exponent.
For equal-base solving, use a positive base that is not $1$ .

2. Solve exactly by using equal bases

Start by checking whether both sides can be written as powers of the same base. For example,

125=5^3

, so

5^{x-1}=125

can be rewritten as

5^{x-1}=5^3

. The bases are now equal, positive, and not

1

, so the exponents must be equal.

After matching the exponents, solve the resulting equation using familiar algebra. From

x-1=3

, add

1

to both sides to get

x=4

. Then substitute this value into the original equation to check it.

Do not equate exponents while the bases differ. For example, in

2^x=4^3

, first rewrite

4^3

using base

2

. Since

4=2^2

, the right side becomes

2^6

. The equation then has matching bases.

(a^m)^n=a^{mn}

Rewrite both sides with a common base before comparing exponents.
Use the power-of-a-power rule when a power is raised to a power.
Check an exact answer in the original equation.

3. Estimate a solution with a graph or table

Sometimes the two sides cannot conveniently be rewritten with familiar equal bases. A graphing tool can show where they have the same value. Graph the left side as one function and the right side as another, using the variable as the horizontal coordinate. The horizontal coordinate at their intersection is the solution.

For example, to solve

3^x=20

, graph

y=3^x

and

y=20

. One graph rises as

x

increases; the other is a horizontal line. Their intersection shows the value of

x

for which the two sides agree. A table can also help: compare values of

3^x

at nearby inputs and narrow down the interval containing the solution.

A graph or table usually gives an approximate solution. Use the approximation symbol and follow the requested rounding. Check the result by substituting it into both sides and seeing whether the values are close. In a context, also check units and restrictions. If a model describes elapsed time after an event, a negative time may not make sense.

f(x)=g(x)

Graph each side as a separate function.
The intersection's horizontal coordinate solves the equation.
Label a rounded answer as an estimate and check it in context.

4. A reliable solving routine

First identify the unknown and the two sides of the equation. Decide whether both sides can be rewritten with the same positive base that is not

1

. If so, rewrite the powers, equate the exponents, and solve the resulting equation.

If equal bases are not practical, graph or tabulate the two sides and estimate where they agree. Finally, check the result in the original equation. For an estimate, verify that the two sides are close. For a word problem, state what the answer represents and include units. Check that the value fits the model's stated domain, meaning the inputs for which the model applies.

Choose equal bases for an exact solution when possible.
Use a graph or table to estimate when a common base is not practical.
Check the equation, units, and relevant domain restrictions.

Worked example

Rewrite a power using the same base

Solve

5^{x-1}=125

.

Write both sides with base 5
Since $125=5^3$ , replace $125$ with $5^3$ . The bases are now equal, positive, and not $1$ .
$5^{x-1}=5^3$
Match the exponents
Equal powers with the same positive base, when that base is not $1$ , have equal exponents. Solve the resulting equation.
$x-1=3\Rightarrow x=4$
Check
Substitute $x=4$ into the original equation. Both sides equal $125$ , so the answer works.
$5^{4-1}=5^3=125$

Answer:

x=4

Check: The original left side becomes

5^3=125

, equal to the right side.

Worked example

Estimate a solution by graphing

Estimate the solution to

3^x=20

to the nearest hundredth.

Intersection of the two sides

A window of the model; the intersection gives the approximate solution.

Graph each side
Graph the two sides as $y=3^x$ and $y=20$ . The solution is the horizontal coordinate where the rising exponential curve meets the horizontal line.
$y=3^x\qquad y=20$
Read the intersection
The intersection is approximately at $x=2.73$ . It is an estimate because the graph provides a decimal reading rather than an exact value.
$x\approx2.73$
Check the estimate
At $x=2.73$ , the left side is about $20.05$ , close to $20$ . This supports the estimate to the requested precision.
$3^{2.73}\approx20.05$

Answer:

x\approx2.73

Check: The left side at the estimate is close to 20. A more precise value is about

2.73

to the nearest hundredth.

Worked example

Solve an exponential model in context

A culture begins with 80 cells and doubles every hour. Its model is

N=80\cdot2^t

, where

N

is the number of cells and

t

is time in hours. After how many hours will the model predict 640 cells?

Set the model equal to the target
The question asks when the model predicts 640 cells, so set its output equal to 640.
$640=80\cdot2^t$
Isolate the power
Divide both sides by 80. This keeps the equation balanced and leaves the exponential expression by itself.
$8=2^t$
Rewrite with matching bases
Since $8=2^3$ , both sides have base $2$ , which is positive and not $1$ . Match the exponents. The resulting time is nonnegative, as required for elapsed time after the culture begins.
$2^3=2^t\Rightarrow t=3$

Answer: The model predicts 640 cells after

3

hours.

Check: At

t=3

,

80\cdot2^3=80\cdot8=640

cells.

Common mistakes and how to avoid them

Equating exponents when the bases are different.

Correction: First rewrite both sides with the same base. For this method, the common base must be positive and not

1

.

Using an exponential base that is zero or negative, or treating a base of

1

as useful for matching exponents.

Correction: Use a positive base other than

1

for the equal-base method. A base of

1

gives the same value for every exponent, so it cannot identify a unique exponent.

Giving a graph estimate as if it were exact.

Correction: Use an approximation symbol, state the requested rounding, and check that both sides are close at the estimate.

Solving the equation but ignoring what the variable represents.

Correction: Check units and restrictions from the situation. For elapsed time after an event, a negative answer may not fit the model.

Lesson summary

An exponential equation has a variable in an exponent.
For exact solving, rewrite both sides using the same positive base that is not $1$ , then equate exponents.
When equal bases are not practical, graph or tabulate both sides to estimate where they agree.
Check the solution in the original equation and make sure it fits the context.

Check your understanding

Question 1

Solve

2^{x+1}=16

.

$x=2$
$x=3$
$x=4$
$x=5$

Show answer and explanation

x=3

Write

16=2^4

. Then

x+1=4

, so

x=3

.

Question 2

A graph shows that

5^x

meets the line

y=30

at about

x=2.11

. What does this value represent?

The approximate value of $5^x$
The approximate exponent that makes $5^x=30$
The exact value of $5^x$ for every $x$
The number of times 30 is multiplied by 5

Show answer and explanation

The approximate exponent that makes

5^x=30

At the intersection, the two outputs agree. Its horizontal coordinate is the approximate exponent that solves the equation.

Question 3

A quantity follows

Q=12\cdot3^t

. Which equation should be solved to find when the quantity reaches 324?

$12+3^t=324$
$3\cdot12^t=324$
$12\cdot3^t=324$
$12\cdot t^3=324$

Show answer and explanation

12\cdot3^t=324

The model is

Q=12\cdot3^t

. Set its output

Q

equal to the target value, 324.

Key terms

Base: The number being raised to a power.
Exponent: The number that tells the power to which the base is raised.
Exponential equation: An equation in which a variable appears in an exponent.
Approximate solution: A value close to the solution, often estimated from a graph or table.
Domain: The set of input values for which a model applies.

Continue through Math 30-2

View the complete Math 30-2 Alberta Grade 12 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation RF5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

RF5 · Solve problems involving exponential equations

Alberta Grade 12 Mathematics

Use equal bases, graphs, and context to find an unknown exponent.

What you will learn

1. Review powers and identify the unknown

2. Solve exactly by using equal bases

3. Estimate a solution with a graph or table

4. A reliable solving routine

Rewrite a power using the same base

Estimate a solution by graphing

Solve an exponential model in context

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-2

About this lesson and its review