Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Feb 06, 2024

Solve the exponential equation $9^{x}=\frac{1}{\sqrt{3}}$ for the value of $x$ .

STEP 1

Assumptions
1. We are given the equation $9^{x} = \frac{1}{\sqrt{3}}$ .
2. We need to solve for the variable $x$ .

STEP 2

We can rewrite the equation using properties of exponents to express both sides with the same base.
$9^{x} = \left(\sqrt{3}\right)^{-1}$

STEP 3

Since $9$ is a power of $3$ , we can rewrite $9$ as $3^2$ .
$\left(3^2\right)^{x} = \left(\sqrt{3}\right)^{-1}$

STEP 4

We can use the property of exponents that states $(a^b)^c = a^{bc}$ to simplify the left side of the equation.
$3^{2x} = \left(\sqrt{3}\right)^{-1}$

STEP 5

Now, we rewrite the square root of $3$ as $3$ raised to the power of $\frac{1}{2}$ .
$3^{2x} = \left(3^{\frac{1}{2}}\right)^{-1}$

STEP 6

We can use the property of exponents that states $(a^b)^c = a^{bc}$ to simplify the right side of the equation.
$3^{2x} = 3^{-\frac{1}{2}}$

STEP 7

Since the bases on both sides of the equation are now the same, we can set the exponents equal to each other.
$2x = -\frac{1}{2}$

STEP 8

To solve for $x$ , we divide both sides of the equation by $2$ .
$x = \frac{-\frac{1}{2}}{2}$

STEP 9

Simplify the right side of the equation by multiplying the numerator by the reciprocal of the denominator.
$x = -\frac{1}{2} \times \frac{1}{2}$

STEP 10

Calculate the value of $x$ .
$x = -\frac{1}{4}$
The solution to the equation $9^{x} = \frac{1}{\sqrt{3}}$ is $x = -\frac{1}{4}$ .

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