Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Simplify $2 \log (8)$ by moving $2$ inside the logarithm.

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

Step 1.4

Raise $8$ to the power of $2$ .

$\log (x^{\frac{1}{2}}) - \log (y^{3}) + \log (64) - \log (t)$

Step 2

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

$\log (\frac{x^{\frac{1}{2}}}{y^{3}}) + \log (64) - \log (t)$

Step 3

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

$\log (\frac{x^{\frac{1}{2}}}{y^{3}} \cdot 64) - \log (t)$

Step 4

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

$\log (\frac{\frac{x^{\frac{1}{2}}}{y^{3}} \cdot 64}{t})$

Step 5

Combine $\frac{x^{\frac{1}{2}}}{y^{3}}$ and $64$ .

$\log (\frac{\frac{x^{\frac{1}{2}} \cdot 64}{y^{3}}}{t})$

Step 6

Move $64$ to the left of $x^{\frac{1}{2}}$ .

$\log (\frac{\frac{64 x^{\frac{1}{2}}}{y^{3}}}{t})$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{64 x^{\frac{1}{2}}}{y^{3}} \cdot \frac{1}{t})$

Step 8

Combine.

$\log (\frac{64 x^{\frac{1}{2}} \cdot 1}{y^{3} t})$

Step 9

Multiply $64$ by $1$ .

$\log (\frac{64 x^{\frac{1}{2}}}{y^{3} t})$