Algebra Examples

$2 (\log (18) - \log (3)) + \frac{1}{2} \log (\frac{1}{16})$

Step 1

Simplify each term.

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Step 1.1

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

$2 \log (\frac{18}{3}) + \frac{1}{2} \cdot \log (\frac{1}{16})$

Step 1.2

Divide $18$ by $3$ .

$2 \log (6) + \frac{1}{2} \cdot \log (\frac{1}{16})$

Step 1.3

Simplify $2 \log (6)$ by moving $2$ inside the logarithm.

$\log (6^{2}) + \frac{1}{2} \cdot \log (\frac{1}{16})$

Step 1.4

Raise $6$ to the power of $2$ .

$\log (36) + \frac{1}{2} \cdot \log (\frac{1}{16})$

Step 1.5

Simplify $\frac{1}{2} \log (\frac{1}{16})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (36) + \log ({(\frac{1}{16})}^{\frac{1}{2}})$

Step 1.6

Apply the product rule to $\frac{1}{16}$ .

$\log (36) + \log (\frac{1^{\frac{1}{2}}}{16^{\frac{1}{2}}})$

Step 1.7

One to any power is one.

$\log (36) + \log (\frac{1}{16^{\frac{1}{2}}})$

Step 1.8

Simplify the denominator.

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Step 1.8.1

Rewrite $16$ as $4^{2}$ .

$\log (36) + \log (\frac{1}{{(4^{2})}^{\frac{1}{2}}})$

Step 1.8.2

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

$\log (36) + \log (\frac{1}{4^{2 (\frac{1}{2})}})$

Step 1.8.3

Cancel the common factor of $2$ .

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Step 1.8.3.1

Cancel the common factor.

$\log (36) + \log (\frac{1}{4^{2 (\frac{1}{2})}})$

Step 1.8.3.2

Rewrite the expression.

$\log (36) + \log (\frac{1}{4^{1}})$

Step 1.8.4

Evaluate the exponent.

$\log (36) + \log (\frac{1}{4})$

Step 2

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

$\log (36 (\frac{1}{4}))$

Step 3

Cancel the common factor of $4$ .

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Step 3.1

Factor $4$ out of $36$ .

$\log (4 (9) \frac{1}{4})$

Step 3.2

Cancel the common factor.

$\log (4 \cdot 9 \frac{1}{4})$

Step 3.3

Rewrite the expression.

$\log (9)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\log (9)$

Decimal Form:

$0.95424250 \dots$