Algebra Examples

$3 \cdot 3^{\frac{x}{5}} = 1$

Step 1

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

$\ln (3 \cdot 3^{\frac{x}{5}}) = \ln (1)$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (3 \cdot 3^{\frac{x}{5}})$ as $\ln (3) + \ln (3^{\frac{x}{5}})$ .

$\ln (3) + \ln (3^{\frac{x}{5}}) = \ln (1)$

Step 2.2

Expand $\ln (3^{\frac{x}{5}})$ by moving $\frac{x}{5}$ outside the logarithm.

$\ln (3) + \frac{x}{5} \ln (3) = \ln (1)$

Step 2.3

Combine $\frac{x}{5}$ and $\ln (3)$ .

$\ln (3) + \frac{x \ln (3)}{5} = \ln (1)$

Step 3

Simplify the right side.

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Step 3.1

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

$\ln (3) + \frac{x \ln (3)}{5} = 0$

Step 4

Reorder $\ln (3)$ and $\frac{x \ln (3)}{5}$ .

$\frac{x \ln (3)}{5} + \ln (3) = 0$

Step 5

Subtract $\ln (3)$ from both sides of the equation.

$\frac{x \ln (3)}{5} = - \ln (3)$

Step 6

Multiply both sides of the equation by $5$ .

$5 \frac{x \ln (3)}{5} = 5 (- \ln (3))$

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Cancel the common factor of $5$ .

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Step 7.1.1.1

Cancel the common factor.

$5 \frac{x \ln (3)}{5} = 5 (- \ln (3))$

Step 7.1.1.2

Rewrite the expression.

$x \ln (3) = 5 (- \ln (3))$

Step 7.2

Simplify the right side.

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Step 7.2.1

Multiply $- 1$ by $5$ .

$x \ln (3) = - 5 \ln (3)$

Step 8

Divide each term in $x \ln (3) = - 5 \ln (3)$ by $\ln (3)$ and simplify.

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Step 8.1

Divide each term in $x \ln (3) = - 5 \ln (3)$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{- 5 \ln (3)}{\ln (3)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{- 5 \ln (3)}{\ln (3)}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5 \ln (3)}{\ln (3)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

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

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Step 8.3.1.1

Cancel the common factor.

$x = \frac{- 5 \ln (3)}{\ln (3)}$

Step 8.3.1.2

Divide $- 5$ by $1$ .

$x = - 5$