Precalculus Examples

$3 \log (5) \cdot 3 + \frac{1}{5} \cdot (\log (5) (r - 4)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1

Simplify each term.

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Step 1.1

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

$\log (5^{3}) \cdot 3 + \frac{1}{5} \cdot (\log (5) (r - 4)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.2

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

$\log (125) \cdot 3 + \frac{1}{5} \cdot (\log (5) (r - 4)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.3

Multiply $\log (125) \cdot 3$ .

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Step 1.3.1

Reorder $\log (125)$ and $3$ .

$3 \cdot \log (125) + \frac{1}{5} \cdot (\log (5) (r - 4)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.3.2

Simplify $3 \cdot \log (125)$ by moving $3$ inside the logarithm.

$\log (125^{3}) + \frac{1}{5} \cdot (\log (5) (r - 4)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.4

Raise $125$ to the power of $3$ .

$\log (1953125) + \frac{1}{5} \cdot (\log (5) (r - 4)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.5

Apply the distributive property.

$\log (1953125) + \frac{1}{5} \cdot (\log (5) r + \log (5) \cdot - 4) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.6

Multiply $\log (5) \cdot - 4$ .

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Step 1.6.1

Reorder $\log (5)$ and $- 4$ .

$\log (1953125) + \frac{1}{5} \cdot (\log (5) r - 4 \cdot \log (5)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.6.2

Simplify $- 4 \cdot \log (5)$ by moving $4$ inside the logarithm.

$\log (1953125) + \frac{1}{5} \cdot (\log (5) r - \log (5^{4})) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.7

Raise $5$ to the power of $4$ .

$\log (1953125) + \frac{1}{5} \cdot (\log (5) r - \log (625)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.8

Apply the distributive property.

$\log (1953125) + \frac{1}{5} (\log (5) r) + \frac{1}{5} (- \log (625)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.9

Multiply $\frac{1}{5} (\log (5) r)$ .

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Step 1.9.1

Reorder $\log (5)$ and $\frac{1}{5}$ .

$\log (1953125) + \frac{1}{5} \log (5) r + \frac{1}{5} (- \log (625)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.9.2

Simplify $\frac{1}{5} \log (5)$ by moving $\frac{1}{5}$ inside the logarithm.

$\log (1953125) + \log (5^{\frac{1}{5}}) r + \frac{1}{5} (- \log (625)) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.10

Simplify $\frac{1}{5} \log (625)$ by moving $\frac{1}{5}$ inside the logarithm.

$\log (1953125) + \log (5^{\frac{1}{5}}) r - \log (625^{\frac{1}{5}}) - \frac{1}{2} \cdot (\log (5) r)$

Step 1.11

Multiply $- \frac{1}{2} (\log (5) r)$ .

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Step 1.11.1

Reorder $\log (5)$ and $\frac{1}{2}$ .

$\log (1953125) + \log (5^{\frac{1}{5}}) r - \log (625^{\frac{1}{5}}) - (\frac{1}{2} \log (5)) r$

Step 1.11.2

Simplify $\frac{1}{2} \log (5)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (1953125) + \log (5^{\frac{1}{5}}) r - \log (625^{\frac{1}{5}}) - \log (5^{\frac{1}{2}}) r$

Step 2

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

$\log (5^{\frac{1}{5}}) r + \log (\frac{1953125}{625^{\frac{1}{5}}}) - \log (5^{\frac{1}{2}}) r$

Step 3

Reorder factors in $\log (5^{\frac{1}{5}}) r + \log (\frac{1953125}{625^{\frac{1}{5}}}) - \log (5^{\frac{1}{2}}) r$ .

$r \log (5^{\frac{1}{5}}) + \log (\frac{1953125}{625^{\frac{1}{5}}}) - r \log (5^{\frac{1}{2}})$