Precalculus Examples

$3 \log (x) + \frac{1}{2} \cdot \log (y) - 2 \log (z) - \frac{2}{3} \cdot \log (8)$

Step 1

Simplify each term.

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Step 1.1

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

$\log (x^{3}) + \frac{1}{2} \cdot \log (y) - 2 \log (z) - \frac{2}{3} \cdot \log (8)$

Step 1.2

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

$\log (x^{3}) + \log (y^{\frac{1}{2}}) - 2 \log (z) - \frac{2}{3} \cdot \log (8)$

Step 1.3

Simplify $- 2 \log (z)$ by moving $2$ inside the logarithm.

$\log (x^{3}) + \log (y^{\frac{1}{2}}) - \log (z^{2}) - \frac{2}{3} \cdot \log (8)$

Step 1.4

Multiply $- \frac{2}{3} \log (8)$ .

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Step 1.4.1

Reorder $\log (8)$ and $\frac{2}{3}$ .

$\log (x^{3}) + \log (y^{\frac{1}{2}}) - \log (z^{2}) - (\frac{2}{3} \log (8))$

Step 1.4.2

Simplify $\frac{2}{3} \log (8)$ by moving $\frac{2}{3}$ inside the logarithm.

$\log (x^{3}) + \log (y^{\frac{1}{2}}) - \log (z^{2}) - \log (8^{\frac{2}{3}})$

Step 1.5

Rewrite $8$ as $2^{3}$ .

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

Step 1.6

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

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

Step 1.7

Cancel the common factor of $3$ .

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Step 1.7.1

Cancel the common factor.

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

Step 1.7.2

Rewrite the expression.

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

Step 1.8

Raise $2$ to the power of $2$ .

$\log (x^{3}) + \log (y^{\frac{1}{2}}) - \log (z^{2}) - \log (4)$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (x^{3} y^{\frac{1}{2}}) - \log (z^{2}) - \log (4)$

Step 3

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

$\log (\frac{x^{3} y^{\frac{1}{2}}}{z^{2}}) - \log (4)$

Step 4

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

$\log (\frac{\frac{x^{3} y^{\frac{1}{2}}}{z^{2}}}{4})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x^{3} y^{\frac{1}{2}}}{z^{2}} \cdot \frac{1}{4})$

Step 6

Combine.

$\log (\frac{x^{3} y^{\frac{1}{2}} \cdot 1}{z^{2} \cdot 4})$

Step 7

Simplify the expression.

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Step 7.1

Multiply $x^{3}$ by $1$ .

$\log (\frac{x^{3} y^{\frac{1}{2}}}{z^{2} \cdot 4})$

Step 7.2

Move $4$ to the left of $z^{2}$ .

$\log (\frac{x^{3} y^{\frac{1}{2}}}{4 z^{2}})$