Precalculus Examples

$3^{x - 7} = \frac{1}{27}$

Step 1

Subtract $\frac{1}{27}$ from both sides of the equation.

$3^{x - 7} - \frac{1}{27} = 0$

Step 2

Simplify $3^{x - 7} - \frac{1}{27}$ .

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Step 2.1

To write $3^{x - 7}$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$3^{x - 7} \cdot \frac{27}{27} - \frac{1}{27} = 0$

Step 2.2

Combine $3^{x - 7}$ and $\frac{27}{27}$ .

$\frac{3^{x - 7} \cdot 27}{27} - \frac{1}{27} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{3^{x - 7} \cdot 27 - 1}{27} = 0$

Step 2.4

Multiply $3^{x - 7} \cdot 27$ .

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Step 2.4.1

Rewrite $27$ as $3^{3}$ .

$\frac{3^{x - 7} \cdot 3^{3} - 1}{27} = 0$

Step 2.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3^{x - 7 + 3} - 1}{27} = 0$

Step 2.4.3

Add $- 7$ and $3$ .

$\frac{3^{x - 4} - 1}{27} = 0$

Step 3

Set the numerator equal to zero.

$3^{x - 4} - 1 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $1$ to both sides of the equation.

$3^{x - 4} = 1$

Step 4.2

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

$\ln (3^{x - 4}) = \ln (1)$

Step 4.3

Expand $\ln (3^{x - 4})$ by moving $x - 4$ outside the logarithm.

$(x - 4) \ln (3) = \ln (1)$

Step 4.4

Simplify the left side.

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Step 4.4.1

Apply the distributive property.

$x \ln (3) - 4 \ln (3) = \ln (1)$

Step 4.5

Simplify the right side.

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Step 4.5.1

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

$x \ln (3) - 4 \ln (3) = 0$

Step 4.6

Add $4 \ln (3)$ to both sides of the equation.

$x \ln (3) = 4 \ln (3)$

Step 4.7

Divide each term in $x \ln (3) = 4 \ln (3)$ by $\ln (3)$ and simplify.

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Step 4.7.1

Divide each term in $x \ln (3) = 4 \ln (3)$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{4 \ln (3)}{\ln (3)}$

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

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

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Step 4.7.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{4 \ln (3)}{\ln (3)}$

Step 4.7.2.1.2

Divide $x$ by $1$ .

$x = \frac{4 \ln (3)}{\ln (3)}$

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

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

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Step 4.7.3.1.1

Cancel the common factor.

$x = \frac{4 \ln (3)}{\ln (3)}$

Step 4.7.3.1.2

Divide $4$ by $1$ .

$x = 4$