Precalculus Examples

$100 {(1.05)}^{x} = 200 {(1.09)}^{3 x}$

Step 1

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

$\ln (100 \cdot {1.05}^{x}) = \ln (200 \cdot {1.09}^{3 x})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (100 \cdot {1.05}^{x})$ as $\ln (100) + \ln ({1.05}^{x})$ .

$\ln (100) + \ln ({1.05}^{x}) = \ln (200 \cdot {1.09}^{3 x})$

Step 2.2

Expand $\ln ({1.05}^{x})$ by moving $x$ outside the logarithm.

$\ln (100) + x \ln (1.05) = \ln (200 \cdot {1.09}^{3 x})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (200 \cdot {1.09}^{3 x})$ as $\ln (200) + \ln ({1.09}^{3 x})$ .

$\ln (100) + x \ln (1.05) = \ln (200) + \ln ({1.09}^{3 x})$

Step 3.2

Expand $\ln ({1.09}^{3 x})$ by moving $3 x$ outside the logarithm.

$\ln (100) + x \ln (1.05) = \ln (200) + 3 x \ln (1.09)$

Step 4

Reorder $\ln (100)$ and $x \ln (1.05)$ .

$x \ln (1.05) + \ln (100) = \ln (200) + 3 x \ln (1.09)$

Step 5

Reorder $\ln (200)$ and $3 x \ln (1.09)$ .

$x \ln (1.05) + \ln (100) = 3 x \ln (1.09) + \ln (200)$

Step 6

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

$x \ln (1.05) + \ln (100) - 3 x \ln (1.09) - \ln (200) = 0$

Step 7

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

$x \ln (1.05) - 3 x \ln (1.09) + \ln (\frac{100}{200}) = 0$

Step 8

Cancel the common factor of $100$ and $200$ .

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Step 8.1

Factor $100$ out of $100$ .

$x \ln (1.05) - 3 x \ln (1.09) + \ln (\frac{100 (1)}{200}) = 0$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $100$ out of $200$ .

$x \ln (1.05) - 3 x \ln (1.09) + \ln (\frac{100 \cdot 1}{100 \cdot 2}) = 0$

Step 8.2.2

Cancel the common factor.

$x \ln (1.05) - 3 x \ln (1.09) + \ln (\frac{100 \cdot 1}{100 \cdot 2}) = 0$

Step 8.2.3

Rewrite the expression.

$x \ln (1.05) - 3 x \ln (1.09) + \ln (\frac{1}{2}) = 0$

Step 9

Subtract $\ln (\frac{1}{2})$ from both sides of the equation.

$x \ln (1.05) - 3 x \ln (1.09) = - \ln (\frac{1}{2})$

Step 10

Factor $x$ out of $x \ln (1.05) - 3 x \ln (1.09)$ .

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Step 10.1

Factor $x$ out of $x \ln (1.05)$ .

$x (\ln (1.05)) - 3 x \ln (1.09) = - \ln (\frac{1}{2})$

Step 10.2

Factor $x$ out of $- 3 x \ln (1.09)$ .

$x (\ln (1.05)) + x (- 3 \ln (1.09)) = - \ln (\frac{1}{2})$

Step 10.3

Factor $x$ out of $x (\ln (1.05)) + x (- 3 \ln (1.09))$ .

$x (\ln (1.05) - 3 \ln (1.09)) = - \ln (\frac{1}{2})$

Step 11

Divide each term in $x (\ln (1.05) - 3 \ln (1.09)) = - \ln (\frac{1}{2})$ by $\ln (1.05) - 3 \ln (1.09)$ and simplify.

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Step 11.1

Divide each term in $x (\ln (1.05) - 3 \ln (1.09)) = - \ln (\frac{1}{2})$ by $\ln (1.05) - 3 \ln (1.09)$ .

$\frac{x (\ln (1.05) - 3 \ln (1.09))}{\ln (1.05) - 3 \ln (1.09)} = \frac{- \ln (\frac{1}{2})}{\ln (1.05) - 3 \ln (1.09)}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $\ln (1.05) - 3 \ln (1.09)$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{x (\ln (1.05) - 3 \ln (1.09))}{\ln (1.05) - 3 \ln (1.09)} = \frac{- \ln (\frac{1}{2})}{\ln (1.05) - 3 \ln (1.09)}$

Step 11.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{1}{2})}{\ln (1.05) - 3 \ln (1.09)}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{1}{2})}{\ln (1.05) - 3 \ln (1.09)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{1}{2})}{\ln (1.05) - 3 \ln (1.09)}$

Decimal Form:

$x = - 3.30474642 \dots$