Finite Math Examples

${(\frac{5}{6})}^{x} = \frac{36}{25}$

Step 1

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

$\ln ({(\frac{5}{6})}^{x}) = \ln (\frac{36}{25})$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{5}{6})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{5}{6}) = \ln (\frac{36}{25})$

Step 2.2

Rewrite $\ln (\frac{5}{6})$ as $\ln (5) - \ln (6)$ .

$x (\ln (5) - \ln (6)) = \ln (\frac{36}{25})$

Step 2.3

Rewrite $\ln (6)$ as $\ln (2 (3))$ .

$x (\ln (5) - \ln (2 (3))) = \ln (\frac{36}{25})$

Step 2.4

Rewrite $\ln (2 (3))$ as $\ln (2) + \ln (3)$ .

$x (\ln (5) - (\ln (2) + \ln (3))) = \ln (\frac{36}{25})$

Step 2.5

Apply the distributive property.

$x (\ln (5) - \ln (2) - \ln (3)) = \ln (\frac{36}{25})$

Step 3

Simplify the left side.

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Step 3.1

Simplify $x (\ln (5) - \ln (2) - \ln (3))$ .

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Step 3.1.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x (\ln (\frac{5}{2}) - \ln (3)) = \ln (\frac{36}{25})$

Step 3.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (\frac{\frac{5}{2}}{3}) = \ln (\frac{36}{25})$

Step 3.1.3

Multiply the numerator by the reciprocal of the denominator.

$x \ln (\frac{5}{2} \cdot \frac{1}{3}) = \ln (\frac{36}{25})$

Step 3.1.4

Multiply $\frac{5}{2} \cdot \frac{1}{3}$ .

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Step 3.1.4.1

Multiply $\frac{5}{2}$ by $\frac{1}{3}$ .

$x \ln (\frac{5}{2 \cdot 3}) = \ln (\frac{36}{25})$

Step 3.1.4.2

Multiply $2$ by $3$ .

$x \ln (\frac{5}{6}) = \ln (\frac{36}{25})$

Step 4

Divide each term in $x \ln (\frac{5}{6}) = \ln (\frac{36}{25})$ by $\ln (\frac{5}{6})$ and simplify.

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Step 4.1

Divide each term in $x \ln (\frac{5}{6}) = \ln (\frac{36}{25})$ by $\ln (\frac{5}{6})$ .

$\frac{x \ln (\frac{5}{6})}{\ln (\frac{5}{6})} = \frac{\ln (\frac{36}{25})}{\ln (\frac{5}{6})}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $\ln (\frac{5}{6})$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{x \ln (\frac{5}{6})}{\ln (\frac{5}{6})} = \frac{\ln (\frac{36}{25})}{\ln (\frac{5}{6})}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{36}{25})}{\ln (\frac{5}{6})}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{36}{25})}{\ln (\frac{5}{6})}$

Decimal Form:

$x = - 2$