Calculus Examples

$10 ({1.06}^{x}) = 12 ({1.14}^{x})$

Step 1

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

$\ln (10 \cdot {1.06}^{x}) = \ln (12 \cdot {1.14}^{x})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (10 \cdot {1.06}^{x})$ as $\ln (10) + \ln ({1.06}^{x})$ .

$\ln (10) + \ln ({1.06}^{x}) = \ln (12 \cdot {1.14}^{x})$

Step 2.2

Expand $\ln ({1.06}^{x})$ by moving $x$ outside the logarithm.

$\ln (10) + x \ln (1.06) = \ln (12 \cdot {1.14}^{x})$

Step 3

Expand the right side.

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Step 3.1

Rewrite $\ln (12 \cdot {1.14}^{x})$ as $\ln (12) + \ln ({1.14}^{x})$ .

$\ln (10) + x \ln (1.06) = \ln (12) + \ln ({1.14}^{x})$

Step 3.2

Expand $\ln ({1.14}^{x})$ by moving $x$ outside the logarithm.

$\ln (10) + x \ln (1.06) = \ln (12) + x \ln (1.14)$

Step 4

Reorder $\ln (10)$ and $x \ln (1.06)$ .

$x \ln (1.06) + \ln (10) = \ln (12) + x \ln (1.14)$

Step 5

Reorder $\ln (12)$ and $x \ln (1.14)$ .

$x \ln (1.06) + \ln (10) = x \ln (1.14) + \ln (12)$

Step 6

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

$x \ln (1.06) + \ln (10) - x \ln (1.14) - \ln (12) = 0$

Step 7

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

$x \ln (1.06) - x \ln (1.14) + \ln (\frac{10}{12}) = 0$

Step 8

Cancel the common factor of $10$ and $12$ .

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Step 8.1

Factor $2$ out of $10$ .

$x \ln (1.06) - x \ln (1.14) + \ln (\frac{2 (5)}{12}) = 0$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $12$ .

$x \ln (1.06) - x \ln (1.14) + \ln (\frac{2 \cdot 5}{2 \cdot 6}) = 0$

Step 8.2.2

Cancel the common factor.

$x \ln (1.06) - x \ln (1.14) + \ln (\frac{2 \cdot 5}{2 \cdot 6}) = 0$

Step 8.2.3

Rewrite the expression.

$x \ln (1.06) - x \ln (1.14) + \ln (\frac{5}{6}) = 0$

Step 9

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

$x \ln (1.06) - x \ln (1.14) = - \ln (\frac{5}{6})$

Step 10

Factor $x$ out of $x \ln (1.06) - x \ln (1.14)$ .

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Step 10.1

Factor $x$ out of $x \ln (1.06)$ .

$x (\ln (1.06)) - x \ln (1.14) = - \ln (\frac{5}{6})$

Step 10.2

Factor $x$ out of $- x \ln (1.14)$ .

$x (\ln (1.06)) + x (- 1 \ln (1.14)) = - \ln (\frac{5}{6})$

Step 10.3

Factor $x$ out of $x (\ln (1.06)) + x (- 1 \ln (1.14))$ .

$x (\ln (1.06) - 1 \ln (1.14)) = - \ln (\frac{5}{6})$

Step 11

Rewrite $- 1 \ln (1.14)$ as $- \ln (1.14)$ .

$x (\ln (1.06) - \ln (1.14)) = - \ln (\frac{5}{6})$

Step 12

Divide each term in $x (\ln (1.06) - \ln (1.14)) = - \ln (\frac{5}{6})$ by $\ln (1.06) - \ln (1.14)$ and simplify.

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Step 12.1

Divide each term in $x (\ln (1.06) - \ln (1.14)) = - \ln (\frac{5}{6})$ by $\ln (1.06) - \ln (1.14)$ .

$\frac{x (\ln (1.06) - \ln (1.14))}{\ln (1.06) - \ln (1.14)} = \frac{- \ln (\frac{5}{6})}{\ln (1.06) - \ln (1.14)}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Cancel the common factor of $\ln (1.06) - \ln (1.14)$ .

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Step 12.2.1.1

Cancel the common factor.

$\frac{x (\ln (1.06) - \ln (1.14))}{\ln (1.06) - \ln (1.14)} = \frac{- \ln (\frac{5}{6})}{\ln (1.06) - \ln (1.14)}$

Step 12.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{5}{6})}{\ln (1.06) - \ln (1.14)}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{5}{6})}{\ln (1.06) - \ln (1.14)}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{5}{6})}{\ln (1.06) - \ln (1.14)}$

Decimal Form:

$x = - 2.50581603 \dots$