Precalculus Examples

$\frac{\log (\sqrt{27}) + \log (8) + \log (\sqrt{100})}{\log (120)}$

Step 1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.1

Factor $9$ out of $27$ .

$\frac{\log (\sqrt{9 (3)}) + \log (8) + \log (\sqrt{100})}{\log (120)}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\log (\sqrt{3^{2} \cdot 3}) + \log (8) + \log (\sqrt{100})}{\log (120)}$

Step 2

Pull terms out from under the radical.

$\frac{\log (3 \sqrt{3}) + \log (8) + \log (\sqrt{100})}{\log (120)}$

Step 3

Rewrite $100$ as $10^{2}$ .

$\frac{\log (3 \sqrt{3}) + \log (8) + \log (\sqrt{10^{2}})}{\log (120)}$

Step 4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\log (3 \sqrt{3}) + \log (8) + \log (10)}{\log (120)}$

Step 5

Logarithm base $10$ of $10$ is $1$ .

$\frac{\log (3 \sqrt{3}) + \log (8) + 1}{\log (120)}$

Step 6

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

$\frac{\log (3 \sqrt{3} \cdot 8) + 1}{\log (120)}$

Step 7

Multiply $8$ by $3$ .

$\frac{\log (24 \sqrt{3}) + 1}{\log (120)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (24 \sqrt{3}) + 1}{\log (120)}$

Decimal Form:

$1.25952072 \dots$