Precalculus Examples

$\frac{\frac{\log (78125)}{\log (5)} - 3}{4}$

Step 1

Simplify the numerator.

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Step 1.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{\log (5)}{\log (5)}$ .

$\frac{\frac{\log (78125)}{\log (5)} - 3 \cdot \frac{\log (5)}{\log (5)}}{4}$

Step 1.2

Combine $- 3$ and $\frac{\log (5)}{\log (5)}$ .

$\frac{\frac{\log (78125)}{\log (5)} + \frac{- 3 \log (5)}{\log (5)}}{4}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{\frac{\log (78125) - 3 \log (5)}{\log (5)}}{4}$

Step 1.4

Rewrite $\frac{\log (78125) - 3 \log (5)}{\log (5)}$ in a factored form.

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Step 1.4.1

Simplify $- 3 \log (5)$ by moving $3$ inside the logarithm.

$\frac{\frac{\log (78125) - \log (5^{3})}{\log (5)}}{4}$

Step 1.4.2

Raise $5$ to the power of $3$ .

$\frac{\frac{\log (78125) - \log (125)}{\log (5)}}{4}$

Step 1.4.3

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

$\frac{\frac{\log (\frac{78125}{125})}{\log (5)}}{4}$

Step 1.5

Cancel the common factor of $78125$ and $125$ .

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Step 1.5.1

Factor $125$ out of $78125$ .

$\frac{\frac{\log (\frac{125 \cdot 625}{125})}{\log (5)}}{4}$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $125$ out of $125$ .

$\frac{\frac{\log (\frac{125 \cdot 625}{125 (1)})}{\log (5)}}{4}$

Step 1.5.2.2

Cancel the common factor.

$\frac{\frac{\log (\frac{125 \cdot 625}{125 \cdot 1})}{\log (5)}}{4}$

Step 1.5.2.3

Rewrite the expression.

$\frac{\frac{\log (\frac{625}{1})}{\log (5)}}{4}$

Step 1.5.2.4

Divide $625$ by $1$ .

$\frac{\frac{\log (625)}{\log (5)}}{4}$

Step 1.6

Rewrite $\log (625)$ as $\log (5^{4})$ .

$\frac{\frac{\log (5^{4})}{\log (5)}}{4}$

Step 1.7

Expand $\log (5^{4})$ by moving $4$ outside the logarithm.

$\frac{\frac{4 \log (5)}{\log (5)}}{4}$

Step 1.8

Cancel the common factor of $\log (5)$ .

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Step 1.8.1

Cancel the common factor.

$\frac{\frac{4 \log (5)}{\log (5)}}{4}$

Step 1.8.2

Divide $4$ by $1$ .

$\frac{4}{4}$

Step 2

Divide $4$ by $4$ .

$1$