Using Like Bases to Solve Exponential Equations

Using the One-to-One Property of Exponential Functions to Solve Exponential Equations

For any algebraic expressions $Sand T$, and any positive real number $b≠1$,

$b^S=b^T$ if and only if $S=T$

How To

Given an exponential equation with the form $b^S=b^T$, where $S$ and $T$ are algebraic expressions with an unknown, solve for the unknown.

1. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form $b^S=b^T$.

2. Use the one-to-one property to set the exponents equal.

3. Solve the resulting equation, $S=T$, for the unknown.

Example 1

Solving an Exponential Equation with a Common Base

Solve $2^{x−1}=2^{2x−4}$.

Solution

$2^{x−1}=2^{2x−4}$	The common base is 2.
$x−1=2x−4$	By the one-to-one property the exponents must be equal.
$x= 3$	Solve for x.

Try It #1

Solve $5^{2x}=5^{3x+2}$.

How To

Given an exponential equation with unlike bases, use the one-to-one property to solve it.

1. Rewrite each side in the equation as a power with a common base.

2. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form $b^S=b^T$.

3. Use the one-to-one property to set the exponents equal.

4. Solve the resulting equation, $S=T$, for the unknown.

Example 2

Solving Equations by Rewriting Them to Have a Common Base

Solve $8^{x+2}=16^{x+1}$.

Solution

Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.

For example, consider the equation $256=4^{x−5}$. We can rewrite both sides of this equation as a power of $2$. Then we apply the rules of exponents, along with the one-to-one property, to solve for $x$:

$8^{x+2}=16^{x+1}$
$(2^3)^{x+2} = (2^4)^{x+1}$	Write $8$ and $16$ as powers of $2$.
$2^{3x+6} = 2^{4x+4}$	To take a power of a power, multiply exponents.
$3x+6 = 4x+4$	Use the one-to-one property to set the exponents equal.
$x=2$	Solve for $x$.

Try It #2

Solve $5^{2x}=25^{3x+2}$.

Example 3

Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base

Solve $2^{5x}=\sqrt{2}$.

Solution

$2^{5x}=2\dfrac{1}{2}$	Write the square root of $2$ as a power of $2$.
$x=\dfrac{1}{2}$	Use the one-to-one property.
$x=\dfrac{1}{10}$	Solve for $x$.

Try It #3

Solve 5x=5–√.

Q&A

Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?

No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined.

Example 4

Solving an Equation with Positive and Negative Powers

Solve $3^{x+1}=−2$.

Solution

This equation has no solution. There is no real value of $x$ that will make the equation a true statement because any power of a positive number is positive.

Analysis

Figure 2 shows that the two graphs do not cross so the left side is never equal to the right side. Thus the equation has no solution.
Graph of 3^(x+1)=-2 and y=-2. The graph notes that they do not cross.

Figure 2

Try It #4

Solve $2^x=−100$.