Basic Math Examples

$200000 = 100000 {(1 + \frac{8}{100})}^{n}$

Step 1

Rewrite the equation as $100000 {(1 + \frac{8}{100})}^{n} = 200000$ .

$100000 {(1 + \frac{8}{100})}^{n} = 200000$

Step 2

Simplify.

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Step 2.1

Reduce the expression $\frac{8}{100}$ by cancelling the common factors.

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Step 2.1.1

Factor $4$ out of $8$ .

$100000 {(1 + \frac{4 (2)}{100})}^{n} = 200000$

Step 2.1.2

Factor $4$ out of $100$ .

$100000 {(1 + \frac{4 \cdot 2}{4 \cdot 25})}^{n} = 200000$

Step 2.1.3

Cancel the common factor.

$100000 {(1 + \frac{4 \cdot 2}{4 \cdot 25})}^{n} = 200000$

Step 2.1.4

Rewrite the expression.

$100000 {(1 + \frac{2}{25})}^{n} = 200000$

Step 2.2

Write $1$ as a fraction with a common denominator.

$100000 {(\frac{25}{25} + \frac{2}{25})}^{n} = 200000$

Step 2.3

Combine the numerators over the common denominator.

$100000 {(\frac{25 + 2}{25})}^{n} = 200000$

Step 2.4

Add $25$ and $2$ .

$100000 {(\frac{27}{25})}^{n} = 200000$

Step 3

Divide each term in $100000 {(\frac{27}{25})}^{n} = 200000$ by $100000$ and simplify.

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Step 3.1

Divide each term in $100000 {(\frac{27}{25})}^{n} = 200000$ by $100000$ .

$\frac{100000 {(\frac{27}{25})}^{n}}{100000} = \frac{200000}{100000}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $100000$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{100000 {(\frac{27}{25})}^{n}}{100000} = \frac{200000}{100000}$

Step 3.2.1.2

Divide ${(\frac{27}{25})}^{n}$ by $1$ .

${(\frac{27}{25})}^{n} = \frac{200000}{100000}$

Step 3.2.2

Apply the product rule to $\frac{27}{25}$ .

$\frac{27^{n}}{25^{n}} = \frac{200000}{100000}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $200000$ by $100000$ .

$\frac{27^{n}}{25^{n}} = 2$

Step 4

Multiply both sides by $25^{n}$ .

$\frac{27^{n}}{25^{n}} \cdot 25^{n} = 2 \cdot 25^{n}$

Step 5

Simplify the left side.

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Step 5.1

Cancel the common factor of $25^{n}$ .

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Step 5.1.1

Cancel the common factor.

$\frac{27^{n}}{25^{n}} \cdot 25^{n} = 2 \cdot 25^{n}$

Step 5.1.2

Rewrite the expression.

$27^{n} = 2 \cdot 25^{n}$

Step 6

Solve for $n$ .

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Step 6.1

Take the log of both sides of the equation.

$\ln (27^{n}) = \ln (2 \cdot 25^{n})$

Step 6.2

Expand $\ln (27^{n})$ by moving $n$ outside the logarithm.

$n \ln (27) = \ln (2 \cdot 25^{n})$

Step 6.3

Rewrite $\ln (2 \cdot 25^{n})$ as $\ln (2) + \ln (25^{n})$ .

$n \ln (27) = \ln (2) + \ln (25^{n})$

Step 6.4

Expand $\ln (25^{n})$ by moving $n$ outside the logarithm.

$n \ln (27) = \ln (2) + n \ln (25)$

Step 6.5

Solve the equation for $n$ .

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Step 6.5.1

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

$n \ln (27) - \ln (2) - n \ln (25) = 0$

Step 6.5.2

Add $\ln (2)$ to both sides of the equation.

$n \ln (27) - n \ln (25) = \ln (2)$

Step 6.5.3

Factor $n$ out of $n \ln (27) - n \ln (25)$ .

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Step 6.5.3.1

Factor $n$ out of $n \ln (27)$ .

$n (\ln (27)) - n \ln (25) = \ln (2)$

Step 6.5.3.2

Factor $n$ out of $- n \ln (25)$ .

$n (\ln (27)) + n (- 1 \ln (25)) = \ln (2)$

Step 6.5.3.3

Factor $n$ out of $n (\ln (27)) + n (- 1 \ln (25))$ .

$n (\ln (27) - 1 \ln (25)) = \ln (2)$

Step 6.5.4

Rewrite $- 1 \ln (25)$ as $- \ln (25)$ .

$n (\ln (27) - \ln (25)) = \ln (2)$

Step 6.5.5

Divide each term in $n (\ln (27) - \ln (25)) = \ln (2)$ by $\ln (27) - \ln (25)$ and simplify.

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Step 6.5.5.1

Divide each term in $n (\ln (27) - \ln (25)) = \ln (2)$ by $\ln (27) - \ln (25)$ .

$\frac{n (\ln (27) - \ln (25))}{\ln (27) - \ln (25)} = \frac{\ln (2)}{\ln (27) - \ln (25)}$

Step 6.5.5.2

Simplify the left side.

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Step 6.5.5.2.1

Cancel the common factor of $\ln (27) - \ln (25)$ .

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Step 6.5.5.2.1.1

Cancel the common factor.

$\frac{n (\ln (27) - \ln (25))}{\ln (27) - \ln (25)} = \frac{\ln (2)}{\ln (27) - \ln (25)}$

Step 6.5.5.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (2)}{\ln (27) - \ln (25)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$n = \frac{\ln (2)}{\ln (27) - \ln (25)}$

Decimal Form:

$n = 9.00646834 \dots$