Basic Math Examples

$y = \frac{20 \ln (3)}{\ln (75) - \ln (45)}$

Step 1

Simplify $20 \ln (3)$ by moving $20$ inside the logarithm.

$y = \frac{\ln (3^{20})}{\ln (75) - \ln (45)}$

Step 2

Simplify the denominator.

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Step 2.1

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

$y = \frac{\ln (3^{20})}{\ln (\frac{75}{45})}$

Step 2.2

Cancel the common factor of $75$ and $45$ .

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Step 2.2.1

Factor $15$ out of $75$ .

$y = \frac{\ln (3^{20})}{\ln (\frac{15 (5)}{45})}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $15$ out of $45$ .

$y = \frac{\ln (3^{20})}{\ln (\frac{15 \cdot 5}{15 \cdot 3})}$

Step 2.2.2.2

Cancel the common factor.

$y = \frac{\ln (3^{20})}{\ln (\frac{15 \cdot 5}{15 \cdot 3})}$

Step 2.2.2.3

Rewrite the expression.

$y = \frac{\ln (3^{20})}{\ln (\frac{5}{3})}$

Step 3

Raise $3$ to the power of $20$ .

$y = \frac{\ln (3486784401)}{\ln (\frac{5}{3})}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\ln (3486784401)}{\ln (\frac{5}{3})}$

Decimal Form:

$y = 43.01320206 \dots$