Finite Math Examples

$5^{x - 1} = 17$

Step 1

Subtract $17$ from both sides of the equation.

$5^{x - 1} - 17 = 0$

Step 2

Add $17$ to both sides of the equation.

$5^{x - 1} = 17$

Step 3

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

$\ln (5^{x - 1}) = \ln (17)$

Step 4

Expand $\ln (5^{x - 1})$ by moving $x - 1$ outside the logarithm.

$(x - 1) \ln (5) = \ln (17)$

Step 5

Simplify the left side.

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Step 5.1

Simplify $(x - 1) \ln (5)$ .

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Step 5.1.1

Apply the distributive property.

$x \ln (5) - 1 \ln (5) = \ln (17)$

Step 5.1.2

Rewrite $- 1 \ln (5)$ as $- \ln (5)$ .

$x \ln (5) - \ln (5) = \ln (17)$

Step 6

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

$x \ln (5) - \ln (5) - \ln (17) = 0$

Step 7

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

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Step 7.1

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

$x \ln (5) - \ln (17) = \ln (5)$

Step 7.2

Add $\ln (17)$ to both sides of the equation.

$x \ln (5) = \ln (5) + \ln (17)$

Step 8

Divide each term in $x \ln (5) = \ln (5) + \ln (17)$ by $\ln (5)$ and simplify.

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Step 8.1

Divide each term in $x \ln (5) = \ln (5) + \ln (17)$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (5)}{\ln (5)} + \frac{\ln (17)}{\ln (5)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $\ln (5)$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (5)}{\ln (5)} + \frac{\ln (17)}{\ln (5)}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (5)}{\ln (5)} + \frac{\ln (17)}{\ln (5)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $\ln (5)$ .

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Step 8.3.1.1

Cancel the common factor.

$x = \frac{\ln (5)}{\ln (5)} + \frac{\ln (17)}{\ln (5)}$

Step 8.3.1.2

Rewrite the expression.

$x = 1 + \frac{\ln (17)}{\ln (5)}$