Precalculus Examples

$(5) (8^{x}) = (6) (11^{x})$

Step 1

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

$\ln (5 \cdot 8^{x}) = \ln (6 \cdot 11^{x})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (5 \cdot 8^{x})$ as $\ln (5) + \ln (8^{x})$ .

$\ln (5) + \ln (8^{x}) = \ln (6 \cdot 11^{x})$

Step 2.2

Expand $\ln (8^{x})$ by moving $x$ outside the logarithm.

$\ln (5) + x \ln (8) = \ln (6 \cdot 11^{x})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (6 \cdot 11^{x})$ as $\ln (6) + \ln (11^{x})$ .

$\ln (5) + x \ln (8) = \ln (6) + \ln (11^{x})$

Step 3.2

Expand $\ln (11^{x})$ by moving $x$ outside the logarithm.

$\ln (5) + x \ln (8) = \ln (6) + x \ln (11)$

Step 4

Reorder $\ln (5)$ and $x \ln (8)$ .

$x \ln (8) + \ln (5) = \ln (6) + x \ln (11)$

Step 5

Reorder $\ln (6)$ and $x \ln (11)$ .

$x \ln (8) + \ln (5) = x \ln (11) + \ln (6)$

Step 6

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

$x \ln (8) + \ln (5) - x \ln (11) - \ln (6) = 0$

Step 7

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

$x \ln (8) - x \ln (11) + \ln (\frac{5}{6}) = 0$

Step 8

Subtract $\ln (\frac{5}{6})$ from both sides of the equation.

$x \ln (8) - x \ln (11) = - \ln (\frac{5}{6})$

Step 9

Factor $x$ out of $x \ln (8) - x \ln (11)$ .

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Step 9.1

Factor $x$ out of $x \ln (8)$ .

$x (\ln (8)) - x \ln (11) = - \ln (\frac{5}{6})$

Step 9.2

Factor $x$ out of $- x \ln (11)$ .

$x (\ln (8)) + x (- 1 \ln (11)) = - \ln (\frac{5}{6})$

Step 9.3

Factor $x$ out of $x (\ln (8)) + x (- 1 \ln (11))$ .

$x (\ln (8) - 1 \ln (11)) = - \ln (\frac{5}{6})$

Step 10

Rewrite $- 1 \ln (11)$ as $- \ln (11)$ .

$x (\ln (8) - \ln (11)) = - \ln (\frac{5}{6})$

Step 11

Divide each term in $x (\ln (8) - \ln (11)) = - \ln (\frac{5}{6})$ by $\ln (8) - \ln (11)$ and simplify.

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Step 11.1

Divide each term in $x (\ln (8) - \ln (11)) = - \ln (\frac{5}{6})$ by $\ln (8) - \ln (11)$ .

$\frac{x (\ln (8) - \ln (11))}{\ln (8) - \ln (11)} = \frac{- \ln (\frac{5}{6})}{\ln (8) - \ln (11)}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $\ln (8) - \ln (11)$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{x (\ln (8) - \ln (11))}{\ln (8) - \ln (11)} = \frac{- \ln (\frac{5}{6})}{\ln (8) - \ln (11)}$

Step 11.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{5}{6})}{\ln (8) - \ln (11)}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{5}{6})}{\ln (8) - \ln (11)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{5}{6})}{\ln (8) - \ln (11)}$

Decimal Form:

$x = - 0.57252133 \dots$