Precalculus Examples

${(\frac{2}{3})}^{x} = \frac{81}{16}$

Step 1

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

$\ln ({(\frac{2}{3})}^{x}) = \ln (\frac{81}{16})$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{2}{3})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{2}{3}) = \ln (\frac{81}{16})$

Step 2.2

Rewrite $\ln (\frac{2}{3})$ as $\ln (2) - \ln (3)$ .

$x (\ln (2) - \ln (3)) = \ln (\frac{81}{16})$

Step 3

Simplify the left side.

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Step 3.1

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

$x \ln (\frac{2}{3}) = \ln (\frac{81}{16})$

Step 4

Divide each term in $x \ln (\frac{2}{3}) = \ln (\frac{81}{16})$ by $\ln (\frac{2}{3})$ and simplify.

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Step 4.1

Divide each term in $x \ln (\frac{2}{3}) = \ln (\frac{81}{16})$ by $\ln (\frac{2}{3})$ .

$\frac{x \ln (\frac{2}{3})}{\ln (\frac{2}{3})} = \frac{\ln (\frac{81}{16})}{\ln (\frac{2}{3})}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $\ln (\frac{2}{3})$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{x \ln (\frac{2}{3})}{\ln (\frac{2}{3})} = \frac{\ln (\frac{81}{16})}{\ln (\frac{2}{3})}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{81}{16})}{\ln (\frac{2}{3})}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{81}{16})}{\ln (\frac{2}{3})}$

Decimal Form:

$x = - 4$