Precalculus Examples

$9^{x} - 3^{x} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

$u^{2} - u = 0$

Step 1.4

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

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Step 1.4.1

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

$u \cdot u - u = 0$

Step 1.4.2

Factor $u$ out of $- u$ .

$u \cdot u + u \cdot - 1 = 0$

Step 1.4.3

Factor $u$ out of $u \cdot u + u \cdot - 1$ .

$u (u - 1) = 0$

Step 1.5

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

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

Step 2

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

$3^{x} = 0$

$3^{x} - 1 = 0$

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 $3^{x} - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

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

$3^{x} - 1 = 0$

Step 4.2

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

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Step 4.2.1

Add $1$ to both sides of the equation.

$3^{x} = 1$

Step 4.2.2

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

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

Step 4.2.3

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

$x \ln (3) = \ln (1)$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

The natural logarithm of $1$ is $0$ .

$x \ln (3) = 0$

Step 4.2.5

Divide each term in $x \ln (3) = 0$ by $\ln (3)$ and simplify.

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Step 4.2.5.1

Divide each term in $x \ln (3) = 0$ by $\ln (3)$ .

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

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.

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

Step 4.2.5.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.5.3

Simplify the right side.

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Step 4.2.5.3.1

Divide $0$ by $\ln (3)$ .

$x = 0$

Step 5

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

$x = 0$