Finite Math Examples

$10 ({1.05}^{x}) = 11 ({1.04}^{x})$

Step 1

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

$\ln (10 \cdot {1.05}^{x}) = \ln (11 \cdot {1.04}^{x})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (10 \cdot {1.05}^{x})$ as $\ln (10) + \ln ({1.05}^{x})$ .

$\ln (10) + \ln ({1.05}^{x}) = \ln (11 \cdot {1.04}^{x})$

Step 2.2

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

$\ln (10) + x \ln (1.05) = \ln (11 \cdot {1.04}^{x})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (11 \cdot {1.04}^{x})$ as $\ln (11) + \ln ({1.04}^{x})$ .

$\ln (10) + x \ln (1.05) = \ln (11) + \ln ({1.04}^{x})$

Step 3.2

Expand $\ln ({1.04}^{x})$ by moving $x$ outside the logarithm.

$\ln (10) + x \ln (1.05) = \ln (11) + x \ln (1.04)$

Step 4

Reorder $\ln (10)$ and $x \ln (1.05)$ .

$x \ln (1.05) + \ln (10) = \ln (11) + x \ln (1.04)$

Step 5

Reorder $\ln (11)$ and $x \ln (1.04)$ .

$x \ln (1.05) + \ln (10) = x \ln (1.04) + \ln (11)$

Step 6

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

$x \ln (1.05) + \ln (10) - x \ln (1.04) - \ln (11) = 0$

Step 7

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

$x \ln (1.05) - x \ln (1.04) + \ln (\frac{10}{11}) = 0$

Step 8

Subtract $\ln (\frac{10}{11})$ from both sides of the equation.

$x \ln (1.05) - x \ln (1.04) = - \ln (\frac{10}{11})$

Step 9

Factor $x$ out of $x \ln (1.05) - x \ln (1.04)$ .

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Step 9.1

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

$x (\ln (1.05)) - x \ln (1.04) = - \ln (\frac{10}{11})$

Step 9.2

Factor $x$ out of $- x \ln (1.04)$ .

$x (\ln (1.05)) + x (- 1 \ln (1.04)) = - \ln (\frac{10}{11})$

Step 9.3

Factor $x$ out of $x (\ln (1.05)) + x (- 1 \ln (1.04))$ .

$x (\ln (1.05) - 1 \ln (1.04)) = - \ln (\frac{10}{11})$

Step 10

Rewrite $- 1 \ln (1.04)$ as $- \ln (1.04)$ .

$x (\ln (1.05) - \ln (1.04)) = - \ln (\frac{10}{11})$

Step 11

Divide each term in $x (\ln (1.05) - \ln (1.04)) = - \ln (\frac{10}{11})$ by $\ln (1.05) - \ln (1.04)$ and simplify.

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Step 11.1

Divide each term in $x (\ln (1.05) - \ln (1.04)) = - \ln (\frac{10}{11})$ by $\ln (1.05) - \ln (1.04)$ .

$\frac{x (\ln (1.05) - \ln (1.04))}{\ln (1.05) - \ln (1.04)} = \frac{- \ln (\frac{10}{11})}{\ln (1.05) - \ln (1.04)}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $\ln (1.05) - \ln (1.04)$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{x (\ln (1.05) - \ln (1.04))}{\ln (1.05) - \ln (1.04)} = \frac{- \ln (\frac{10}{11})}{\ln (1.05) - \ln (1.04)}$

Step 11.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{10}{11})}{\ln (1.05) - \ln (1.04)}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{10}{11})}{\ln (1.05) - \ln (1.04)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{10}{11})}{\ln (1.05) - \ln (1.04)}$

Decimal Form:

$x = 9.95983778 \dots$