Precalculus Examples

$8 (9^{x}) - 3^{x} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $9^{x}$ as ${(3^{2})}^{x}$ .

$8 \cdot {(3^{2})}^{x} - 3^{x} = 0$

Step 1.2

Rewrite ${(3^{2})}^{x}$ as ${(3^{x})}^{2}$ .

$8 \cdot {(3^{x})}^{2} - 3^{x} = 0$

Step 1.3

Let $u = 3^{x}$ . Substitute $u$ for all occurrences of $3^{x}$ .

$8 u^{2} - u = 0$

Step 1.4

Factor $u$ out of $8 u^{2} - u$ .

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Step 1.4.1

Factor $u$ out of $8 u^{2}$ .

$u (8 u) - u = 0$

Step 1.4.2

Factor $u$ out of $- u$ .

$u (8 u) + u \cdot - 1 = 0$

Step 1.4.3

Factor $u$ out of $u (8 u) + u \cdot - 1$ .

$u (8 u - 1) = 0$

Step 1.5

Replace all occurrences of $u$ with $3^{x}$ .

$3^{x} (8 \cdot 3^{x} - 1) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3^{x} = 0$

$8 \cdot 3^{x} - 1 = 0$

Step 3

Set $3^{x}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3^{x}$ equal to $0$ .

$3^{x} = 0$

Step 3.2

Solve $3^{x} = 0$ for $x$ .

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Step 3.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (3^{x}) = \ln (0)$

Step 3.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 3.2.3

There is no solution for $3^{x} = 0$

No solution

Step 4

Set $8 \cdot 3^{x} - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $8 \cdot 3^{x} - 1$ equal to $0$ .

$8 \cdot 3^{x} - 1 = 0$

Step 4.2

Solve $8 \cdot 3^{x} - 1 = 0$ for $x$ .

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Step 4.2.1

Add $1$ to both sides of the equation.

$8 \cdot 3^{x} = 1$

Step 4.2.2

Divide each term in $8 \cdot 3^{x} = 1$ by $8$ and simplify.

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Step 4.2.2.1

Divide each term in $8 \cdot 3^{x} = 1$ by $8$ .

$\frac{8 \cdot 3^{x}}{8} = \frac{1}{8}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $8$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{8 \cdot 3^{x}}{8} = \frac{1}{8}$

Step 4.2.2.2.1.2

Divide $3^{x}$ by $1$ .

$3^{x} = \frac{1}{8}$

Step 4.2.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (3^{x}) = \ln (\frac{1}{8})$

Step 4.2.4

Expand $\ln (3^{x})$ by moving $x$ outside the logarithm.

$x \ln (3) = \ln (\frac{1}{8})$

Step 4.2.5

Divide each term in $x \ln (3) = \ln (\frac{1}{8})$ by $\ln (3)$ and simplify.

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Step 4.2.5.1

Divide each term in $x \ln (3) = \ln (\frac{1}{8})$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (\frac{1}{8})}{\ln (3)}$

Step 4.2.5.2

Simplify the left side.

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Step 4.2.5.2.1

Cancel the common factor of $\ln (3)$ .

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Step 4.2.5.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (\frac{1}{8})}{\ln (3)}$

Step 4.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{8})}{\ln (3)}$

Step 5

The final solution is all the values that make $3^{x} (8 \cdot 3^{x} - 1) = 0$ true.

$x = \frac{\ln (\frac{1}{8})}{\ln (3)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{1}{8})}{\ln (3)}$

Decimal Form:

$x = - 1.89278926 \dots$