Basic Math Examples

$100000 = 75000 {(1.095)}^{p}$

Step 1

Rewrite the equation as $75000 \cdot {1.095}^{p} = 100000$ .

$75000 \cdot {1.095}^{p} = 100000$

Step 2

Divide each term in $75000 \cdot {1.095}^{p} = 100000$ by $75000$ and simplify.

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Step 2.1

Divide each term in $75000 \cdot {1.095}^{p} = 100000$ by $75000$ .

$\frac{75000 \cdot {1.095}^{p}}{75000} = \frac{100000}{75000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $75000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{75000 \cdot {1.095}^{p}}{75000} = \frac{100000}{75000}$

Step 2.2.1.2

Divide ${1.095}^{p}$ by $1$ .

${1.095}^{p} = \frac{100000}{75000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $100000$ and $75000$ .

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Step 2.3.1.1

Factor $25000$ out of $100000$ .

${1.095}^{p} = \frac{25000 (4)}{75000}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $25000$ out of $75000$ .

${1.095}^{p} = \frac{25000 \cdot 4}{25000 \cdot 3}$

Step 2.3.1.2.2

Cancel the common factor.

${1.095}^{p} = \frac{25000 \cdot 4}{25000 \cdot 3}$

Step 2.3.1.2.3

Rewrite the expression.

${1.095}^{p} = \frac{4}{3}$

Step 3

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

$\ln ({1.095}^{p}) = \ln (\frac{4}{3})$

Step 4

Expand $\ln ({1.095}^{p})$ by moving $p$ outside the logarithm.

$p \ln (1.095) = \ln (\frac{4}{3})$

Step 5

Divide each term in $p \ln (1.095) = \ln (\frac{4}{3})$ by $\ln (1.095)$ and simplify.

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Step 5.1

Divide each term in $p \ln (1.095) = \ln (\frac{4}{3})$ by $\ln (1.095)$ .

$\frac{p \ln (1.095)}{\ln (1.095)} = \frac{\ln (\frac{4}{3})}{\ln (1.095)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $\ln (1.095)$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{p \ln (1.095)}{\ln (1.095)} = \frac{\ln (\frac{4}{3})}{\ln (1.095)}$

Step 5.2.1.2

Divide $p$ by $1$ .

$p = \frac{\ln (\frac{4}{3})}{\ln (1.095)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$p = \frac{\ln (\frac{4}{3})}{\ln (1.095)}$

Decimal Form:

$p = 3.16989797 \dots$