Precalculus Examples

$3 \log (5 x^{2}) - \frac{1}{2} \cdot \log (y) - \log (7 z)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (5 x^{2})$ by moving $3$ inside the logarithm.

$\log ({(5 x^{2})}^{3}) - \frac{1}{2} \cdot \log (y) - \log (7 z)$

Step 1.2

Apply the product rule to $5 x^{2}$ .

$\log (5^{3} {(x^{2})}^{3}) - \frac{1}{2} \cdot \log (y) - \log (7 z)$

Step 1.3

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

$\log (125 {(x^{2})}^{3}) - \frac{1}{2} \cdot \log (y) - \log (7 z)$

Step 1.4

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (125 x^{2 \cdot 3}) - \frac{1}{2} \cdot \log (y) - \log (7 z)$

Step 1.4.2

Multiply $2$ by $3$ .

$\log (125 x^{6}) - \frac{1}{2} \cdot \log (y) - \log (7 z)$

Step 1.5

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

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Step 1.5.1

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

$\log (125 x^{6}) - (\frac{1}{2} \log (y)) - \log (7 z)$

Step 1.5.2

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

$\log (125 x^{6}) - \log (y^{\frac{1}{2}}) - \log (7 z)$

Step 2

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

$\log (\frac{125 x^{6}}{y^{\frac{1}{2}}}) - \log (7 z)$

Step 3

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

$\log (\frac{\frac{125 x^{6}}{y^{\frac{1}{2}}}}{7 z})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{125 x^{6}}{y^{\frac{1}{2}}} \cdot \frac{1}{7 z})$

Step 5

Combine.

$\log (\frac{125 x^{6} \cdot 1}{y^{\frac{1}{2}} (7 z)})$

Step 6

Simplify the expression.

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Step 6.1

Multiply $125$ by $1$ .

$\log (\frac{125 x^{6}}{y^{\frac{1}{2}} \cdot 7 z})$

Step 6.2

Move $7$ to the left of $y^{\frac{1}{2}}$ .

$\log (\frac{125 x^{6}}{7 \cdot y^{\frac{1}{2}} z})$

Step 6.3

Reorder factors in $\log (\frac{125 x^{6}}{7 y^{\frac{1}{2}} z})$ .

$\log (\frac{125 x^{6}}{7 z y^{\frac{1}{2}}})$