Precalculus Examples

${(\frac{4}{5})}^{6 x + 1} = \frac{5}{4}$

Step 1

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

$\ln ({(\frac{4}{5})}^{6 x + 1}) = \ln (\frac{5}{4})$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{4}{5})}^{6 x + 1})$ by moving $6 x + 1$ outside the logarithm.

$(6 x + 1) \ln (\frac{4}{5}) = \ln (\frac{5}{4})$

Step 2.2

Rewrite $\ln (\frac{4}{5})$ as $\ln (4) - \ln (5)$ .

$(6 x + 1) (\ln (4) - \ln (5)) = \ln (\frac{5}{4})$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(6 x + 1) (\ln (4) - \ln (5))$ .

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Step 3.1.1

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

$(6 x + 1) \ln (\frac{4}{5}) = \ln (\frac{5}{4})$

Step 3.1.2

Apply the distributive property.

$6 x \ln (\frac{4}{5}) + 1 \ln (\frac{4}{5}) = \ln (\frac{5}{4})$

Step 3.1.3

Multiply $\ln (\frac{4}{5})$ by $1$ .

$6 x \ln (\frac{4}{5}) + \ln (\frac{4}{5}) = \ln (\frac{5}{4})$

Step 4

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

$6 x \ln (\frac{4}{5}) + \ln (\frac{4}{5}) - \ln (\frac{5}{4}) = 0$

Step 5

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

$6 x \ln (\frac{4}{5}) + \ln (\frac{\frac{4}{5}}{\frac{5}{4}}) = 0$

Step 6

Simplify each term.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$6 x \ln (\frac{4}{5}) + \ln (\frac{4}{5} \cdot \frac{4}{5}) = 0$

Step 6.2

Multiply $\frac{4}{5} \cdot \frac{4}{5}$ .

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Step 6.2.1

Multiply $\frac{4}{5}$ by $\frac{4}{5}$ .

$6 x \ln (\frac{4}{5}) + \ln (\frac{4 \cdot 4}{5 \cdot 5}) = 0$

Step 6.2.2

Multiply $4$ by $4$ .

$6 x \ln (\frac{4}{5}) + \ln (\frac{16}{5 \cdot 5}) = 0$

Step 6.2.3

Multiply $5$ by $5$ .

$6 x \ln (\frac{4}{5}) + \ln (\frac{16}{25}) = 0$

Step 7

Subtract $\ln (\frac{16}{25})$ from both sides of the equation.

$6 x \ln (\frac{4}{5}) = - \ln (\frac{16}{25})$

Step 8

Divide each term in $6 x \ln (\frac{4}{5}) = - \ln (\frac{16}{25})$ by $6 \ln (\frac{4}{5})$ and simplify.

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Step 8.1

Divide each term in $6 x \ln (\frac{4}{5}) = - \ln (\frac{16}{25})$ by $6 \ln (\frac{4}{5})$ .

$\frac{6 x \ln (\frac{4}{5})}{6 \ln (\frac{4}{5})} = \frac{- \ln (\frac{16}{25})}{6 \ln (\frac{4}{5})}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $6$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{6 x \ln (\frac{4}{5})}{6 \ln (\frac{4}{5})} = \frac{- \ln (\frac{16}{25})}{6 \ln (\frac{4}{5})}$

Step 8.2.1.2

Rewrite the expression.

$\frac{x \ln (\frac{4}{5})}{\ln (\frac{4}{5})} = \frac{- \ln (\frac{16}{25})}{6 \ln (\frac{4}{5})}$

Step 8.2.2

Cancel the common factor of $\ln (\frac{4}{5})$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{x \ln (\frac{4}{5})}{\ln (\frac{4}{5})} = \frac{- \ln (\frac{16}{25})}{6 \ln (\frac{4}{5})}$

Step 8.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{16}{25})}{6 \ln (\frac{4}{5})}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{16}{25})}{6 \ln (\frac{4}{5})}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{16}{25})}{6 \ln (\frac{4}{5})}$

Decimal Form:

$x = - 0.\overline{3}$