Chapter 11, Problem 4CR

To determine

The solution of the equation $3^x=9^{x+1}$.

Answer to Problem 4CR

Solution:

The solution of the equation $3^x=9^{x+1}$ is $x=-2$.

Explanation of Solution

Given information:

The equation $3^x=9^{x+1}$

Explanation:

Consider the equation $3^x=9^{x+1}$.

The number $9$ on right side, can be written as $3^2$.

$\Rightarrow 3^x={\left(3^2\right)}^{x+1}$

By using ${\left(a^m\right)}^n=a^{mn}$,

$\Rightarrow 3^x=3^{2\left(x+1\right)}$

If $a^M=a^N$ implies that $M=N$, by using this property,

$\Rightarrow x=2\left(x+1\right)$

Simplifying right side,

$\Rightarrow x=2x+2$

Subtracting $2x$ from both sides,

$\Rightarrow -x=2$

Multiplying both sides by $-1$,

$\Rightarrow x=-2$

The solution of the equation $3^x=9^{x+1}$ is $x=-2$.