Precalculus Examples

$8 \log (x) + 3 \log (4) - \log (n) - \frac{1}{2} \cdot \log (b)$

Step 1

Simplify each term.

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Step 1.1

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

$\log (x^{8}) + 3 \log (4) - \log (n) - \frac{1}{2} \cdot \log (b)$

Step 1.2

Simplify $3 \log (4)$ by moving $3$ inside the logarithm.

$\log (x^{8}) + \log (4^{3}) - \log (n) - \frac{1}{2} \cdot \log (b)$

Step 1.3

Raise $4$ to the power of $3$ .

$\log (x^{8}) + \log (64) - \log (n) - \frac{1}{2} \cdot \log (b)$

Step 1.4

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

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Step 1.4.1

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

$\log (x^{8}) + \log (64) - \log (n) - (\frac{1}{2} \log (b))$

Step 1.4.2

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

$\log (x^{8}) + \log (64) - \log (n) - \log (b^{\frac{1}{2}})$

Step 2

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

$\log (x^{8} \cdot 64) - \log (n) - \log (b^{\frac{1}{2}})$

Step 3

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

$\log (\frac{x^{8} \cdot 64}{n}) - \log (b^{\frac{1}{2}})$

Step 4

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

$\log (\frac{\frac{x^{8} \cdot 64}{n}}{b^{\frac{1}{2}}})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x^{8} \cdot 64}{n} \cdot \frac{1}{b^{\frac{1}{2}}})$

Step 6

Combine.

$\log (\frac{x^{8} \cdot 64 \cdot 1}{n b^{\frac{1}{2}}})$

Step 7

Multiply $x^{8}$ by $1$ .

$\log (\frac{64 x^{8}}{n b^{\frac{1}{2}}})$