Examples

$3^{x - 1} = 5$

Step 1

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

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

Step 2

Expand $\ln (3^{x - 1})$ by moving $x - 1$ outside the logarithm.

$(x - 1) \ln (3) = \ln (5)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(x - 1) \ln (3)$ .

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Step 3.1.1

Apply the distributive property.

$x \ln (3) - 1 \ln (3) = \ln (5)$

Step 3.1.2

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

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

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (3) - \ln (3) - \ln (5) = 0$

Step 5

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1

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

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

Step 5.2

Add $\ln (5)$ to both sides of the equation.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

Cancel the common factor.

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

Step 6.3.1.2

Rewrite the expression.

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

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