Finite Math Examples

$\ln (0.7) = \ln ({(1.015)}^{- n})$

Step 1

Subtract $\ln ({(1.015)}^{- n})$ from both sides of the equation.

$\ln (0.7) - \ln ({(1.015)}^{- n}) = 0$

Step 2

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

$\ln (\frac{0.7}{{1.015}^{- n}}) = 0$

Step 3

Expand $\ln (\frac{0.7}{{1.015}^{- n}})$ .

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Step 3.1

Rewrite $\ln (\frac{0.7}{{1.015}^{- n}})$ as $\ln (0.7) - \ln ({1.015}^{- n})$ .

$\ln (0.7) - \ln ({1.015}^{- n})$

Step 3.2

Expand $\ln ({1.015}^{- n})$ by moving $- n$ outside the logarithm.

$\ln (0.7) - (- n \ln (1.015))$

Step 3.3

Multiply $- 1$ by $- 1$ .

$\ln (0.7) + 1 (n \ln (1.015))$

Step 3.4

Multiply $n$ by $1$ .

$\ln (0.7) + n \ln (1.015)$

Step 4

The expanded equation is $\ln (0.7) + n \ln (1.015) = 0$ .

$\ln (0.7) + n \ln (1.015) = 0$

Step 5

Reorder $\ln (0.7)$ and $n \ln (1.015)$ .

$n \ln (1.015) + \ln (0.7) = 0$

Step 6

Subtract $\ln (0.7)$ from both sides of the equation.

$n \ln (1.015) = - \ln (0.7)$

Step 7

Divide each term in $n \ln (1.015) = - \ln (0.7)$ by $\ln (1.015)$ and simplify.

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Step 7.1

Divide each term in $n \ln (1.015) = - \ln (0.7)$ by $\ln (1.015)$ .

$\frac{n \ln (1.015)}{\ln (1.015)} = \frac{- \ln (0.7)}{\ln (1.015)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (1.015)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{n \ln (1.015)}{\ln (1.015)} = \frac{- \ln (0.7)}{\ln (1.015)}$

Step 7.2.1.2

Divide $n$ by $1$ .

$n = \frac{- \ln (0.7)}{\ln (1.015)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$n = - \frac{\ln (0.7)}{\ln (1.015)}$