Precalculus Examples

$20 {(1.07)}^{x} = 8 {(1.13)}^{x}$

Step 1

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

$\ln (20 \cdot {1.07}^{x}) = \ln (8 \cdot {1.13}^{x})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (20 \cdot {1.07}^{x})$ as $\ln (20) + \ln ({1.07}^{x})$ .

$\ln (20) + \ln ({1.07}^{x}) = \ln (8 \cdot {1.13}^{x})$

Step 2.2

Expand $\ln ({1.07}^{x})$ by moving $x$ outside the logarithm.

$\ln (20) + x \ln (1.07) = \ln (8 \cdot {1.13}^{x})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (8 \cdot {1.13}^{x})$ as $\ln (8) + \ln ({1.13}^{x})$ .

$\ln (20) + x \ln (1.07) = \ln (8) + \ln ({1.13}^{x})$

Step 3.2

Expand $\ln ({1.13}^{x})$ by moving $x$ outside the logarithm.

$\ln (20) + x \ln (1.07) = \ln (8) + x \ln (1.13)$

Step 4

Reorder $\ln (20)$ and $x \ln (1.07)$ .

$x \ln (1.07) + \ln (20) = \ln (8) + x \ln (1.13)$

Step 5

Reorder $\ln (8)$ and $x \ln (1.13)$ .

$x \ln (1.07) + \ln (20) = x \ln (1.13) + \ln (8)$

Step 6

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

$x \ln (1.07) + \ln (20) - x \ln (1.13) - \ln (8) = 0$

Step 7

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

$x \ln (1.07) - x \ln (1.13) + \ln (\frac{20}{8}) = 0$

Step 8

Cancel the common factor of $20$ and $8$ .

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Step 8.1

Factor $4$ out of $20$ .

$x \ln (1.07) - x \ln (1.13) + \ln (\frac{4 (5)}{8}) = 0$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $4$ out of $8$ .

$x \ln (1.07) - x \ln (1.13) + \ln (\frac{4 \cdot 5}{4 \cdot 2}) = 0$

Step 8.2.2

Cancel the common factor.

$x \ln (1.07) - x \ln (1.13) + \ln (\frac{4 \cdot 5}{4 \cdot 2}) = 0$

Step 8.2.3

Rewrite the expression.

$x \ln (1.07) - x \ln (1.13) + \ln (\frac{5}{2}) = 0$

Step 9

Subtract $\ln (\frac{5}{2})$ from both sides of the equation.

$x \ln (1.07) - x \ln (1.13) = - \ln (\frac{5}{2})$

Step 10

Factor $x$ out of $x \ln (1.07) - x \ln (1.13)$ .

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Step 10.1

Factor $x$ out of $x \ln (1.07)$ .

$x (\ln (1.07)) - x \ln (1.13) = - \ln (\frac{5}{2})$

Step 10.2

Factor $x$ out of $- x \ln (1.13)$ .

$x (\ln (1.07)) + x (- 1 \ln (1.13)) = - \ln (\frac{5}{2})$

Step 10.3

Factor $x$ out of $x (\ln (1.07)) + x (- 1 \ln (1.13))$ .

$x (\ln (1.07) - 1 \ln (1.13)) = - \ln (\frac{5}{2})$

Step 11

Rewrite $- 1 \ln (1.13)$ as $- \ln (1.13)$ .

$x (\ln (1.07) - \ln (1.13)) = - \ln (\frac{5}{2})$

Step 12

Divide each term in $x (\ln (1.07) - \ln (1.13)) = - \ln (\frac{5}{2})$ by $\ln (1.07) - \ln (1.13)$ and simplify.

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Step 12.1

Divide each term in $x (\ln (1.07) - \ln (1.13)) = - \ln (\frac{5}{2})$ by $\ln (1.07) - \ln (1.13)$ .

$\frac{x (\ln (1.07) - \ln (1.13))}{\ln (1.07) - \ln (1.13)} = \frac{- \ln (\frac{5}{2})}{\ln (1.07) - \ln (1.13)}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Cancel the common factor of $\ln (1.07) - \ln (1.13)$ .

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Step 12.2.1.1

Cancel the common factor.

$\frac{x (\ln (1.07) - \ln (1.13))}{\ln (1.07) - \ln (1.13)} = \frac{- \ln (\frac{5}{2})}{\ln (1.07) - \ln (1.13)}$

Step 12.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{5}{2})}{\ln (1.07) - \ln (1.13)}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{5}{2})}{\ln (1.07) - \ln (1.13)}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{5}{2})}{\ln (1.07) - \ln (1.13)}$

Decimal Form:

$x = 16.79449763 \dots$