Precalculus Examples

$\log_{b} (\frac{\sqrt{L \cdot M}}{N^{3}})$

Step 1

Rewrite $\log_{b} (\frac{\sqrt{L \cdot M}}{N^{3}})$ as $\log_{b} (\sqrt{L \cdot M}) - \log_{b} (N^{3})$ .

$\log_{b} (\sqrt{L \cdot M}) - \log_{b} (N^{3})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{L \cdot M}$ as ${(L \cdot M)}^{\frac{1}{2}}$ .

$\log_{b} ({(L \cdot M)}^{\frac{1}{2}}) - \log_{b} (N^{3})$

Step 3

Expand $\log_{b} ({(L \cdot M)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{b} (L \cdot M) - \log_{b} (N^{3})$

Step 4

Expand $\log_{b} (N^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{2} \log_{b} (L \cdot M) - (3 \log_{b} (N))$

Step 5

Multiply $3$ by $- 1$ .

$\frac{1}{2} \log_{b} (L \cdot M) - 3 \log_{b} (N)$

Step 6

Rewrite $\log_{b} (L \cdot M)$ as $\log_{b} (L) + \log_{b} (M)$ .

$\frac{1}{2} (\log_{b} (L) + \log_{b} (M)) - 3 \log_{b} (N)$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{1}{2} \log_{b} (L) + \frac{1}{2} \log_{b} (M) - 3 \log_{b} (N)$

Step 7.2

Combine $\frac{1}{2}$ and $\log_{b} (L)$ .

$\frac{\log_{b} (L)}{2} + \frac{1}{2} \log_{b} (M) - 3 \log_{b} (N)$

Step 7.3

Combine $\frac{1}{2}$ and $\log_{b} (M)$ .

$\frac{\log_{b} (L)}{2} + \frac{\log_{b} (M)}{2} - 3 \log_{b} (N)$