Algebra Examples

{(\frac{4}{3})}^{x} = (\frac{27}{64})

${(\frac{4}{3})}^{x} = (\frac{27}{64})$

Step 1

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

\ln ({(\frac{4}{3})}^{x}) = \ln (\frac{27}{64})

$\ln ({(\frac{4}{3})}^{x}) = \ln (\frac{27}{64})$

Step 2

Expand the left side.

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Step 2.1

Expand

\ln ({(\frac{4}{3})}^{x})

$\ln ({(\frac{4}{3})}^{x})$ by moving

x

$x$ outside the logarithm.

x \ln (\frac{4}{3}) = \ln (\frac{27}{64})

$x \ln (\frac{4}{3}) = \ln (\frac{27}{64})$

Step 2.2

Rewrite

\ln (\frac{4}{3})

$\ln (\frac{4}{3})$ as

\ln (4) - \ln (3)

$\ln (4) - \ln (3)$ .

x (\ln (4) - \ln (3)) = \ln (\frac{27}{64})

$x (\ln (4) - \ln (3)) = \ln (\frac{27}{64})$

x (\ln (4) - \ln (3)) = \ln (\frac{27}{64})

$x (\ln (4) - \ln (3)) = \ln (\frac{27}{64})$

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})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

x \ln (\frac{4}{3}) = \ln (\frac{27}{64})

$x \ln (\frac{4}{3}) = \ln (\frac{27}{64})$

x \ln (\frac{4}{3}) = \ln (\frac{27}{64})

$x \ln (\frac{4}{3}) = \ln (\frac{27}{64})$

Step 4

Divide each term in

x \ln (\frac{4}{3}) = \ln (\frac{27}{64})

$x \ln (\frac{4}{3}) = \ln (\frac{27}{64})$ by

\ln (\frac{4}{3})

$\ln (\frac{4}{3})$ and simplify.

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Step 4.1

Divide each term in

x \ln (\frac{4}{3}) = \ln (\frac{27}{64})

$x \ln (\frac{4}{3}) = \ln (\frac{27}{64})$ by

\ln (\frac{4}{3})

$\ln (\frac{4}{3})$ .

\frac{x \ln (\frac{4}{3})}{\ln (\frac{4}{3})} = \frac{\ln (\frac{27}{64})}{\ln (\frac{4}{3})}

$\frac{x \ln (\frac{4}{3})}{\ln (\frac{4}{3})} = \frac{\ln (\frac{27}{64})}{\ln (\frac{4}{3})}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

\ln (\frac{4}{3})

$\ln (\frac{4}{3})$ .

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Step 4.2.1.1

Cancel the common factor.

\frac{x \ln (\frac{4}{3})}{\ln (\frac{4}{3})} = \frac{\ln (\frac{27}{64})}{\ln (\frac{4}{3})}

$\frac{x \ln (\frac{4}{3})}{\ln (\frac{4}{3})} = \frac{\ln (\frac{27}{64})}{\ln (\frac{4}{3})}$

Step 4.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{\ln (\frac{27}{64})}{\ln (\frac{4}{3})}