Precalculus Examples

$5 (9^{5 x}) = 1$

Step 1

Divide each term in $5 \cdot 9^{5 x} = 1$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 \cdot 9^{5 x} = 1$ by $5$ .

$\frac{5 \cdot 9^{5 x}}{5} = \frac{1}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 \cdot 9^{5 x}}{5} = \frac{1}{5}$

Step 1.2.1.2

Divide $9^{5 x}$ by $1$ .

$9^{5 x} = \frac{1}{5}$

Step 2

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

$\ln (9^{5 x}) = \ln (\frac{1}{5})$

Step 3

Expand $\ln (9^{5 x})$ by moving $5 x$ outside the logarithm.

$5 x \ln (9) = \ln (\frac{1}{5})$

Step 4

Divide each term in $5 x \ln (9) = \ln (\frac{1}{5})$ by $5 \ln (9)$ and simplify.

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Step 4.1

Divide each term in $5 x \ln (9) = \ln (\frac{1}{5})$ by $5 \ln (9)$ .

$\frac{5 x \ln (9)}{5 \ln (9)} = \frac{\ln (\frac{1}{5})}{5 \ln (9)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{5 x \ln (9)}{5 \ln (9)} = \frac{\ln (\frac{1}{5})}{5 \ln (9)}$

Step 4.2.1.2

Rewrite the expression.

$\frac{x \ln (9)}{\ln (9)} = \frac{\ln (\frac{1}{5})}{5 \ln (9)}$

Step 4.2.2

Cancel the common factor of $\ln (9)$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{x \ln (9)}{\ln (9)} = \frac{\ln (\frac{1}{5})}{5 \ln (9)}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{5})}{5 \ln (9)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{1}{5})}{5 \ln (9)}$

Decimal Form:

$x = - 0.14649735 \dots$