Precalculus Examples

$\frac{1}{2} \log (64) - 3 \log (2) + \log (100)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log (64)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (64^{\frac{1}{2}}) - 3 \log (2) + \log (100)$

Step 1.2

Rewrite $64$ as $8^{2}$ .

$\log ({(8^{2})}^{\frac{1}{2}}) - 3 \log (2) + \log (100)$

Step 1.3

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

$\log (8^{2 (\frac{1}{2})}) - 3 \log (2) + \log (100)$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\log (8^{2 (\frac{1}{2})}) - 3 \log (2) + \log (100)$

Step 1.4.2

Rewrite the expression.

$\log (8^{1}) - 3 \log (2) + \log (100)$

Step 1.5

Evaluate the exponent.

$\log (8) - 3 \log (2) + \log (100)$

Step 1.6

Simplify $- 3 \log (2)$ by moving $3$ inside the logarithm.

$\log (8) - \log (2^{3}) + \log (100)$

Step 1.7

Raise $2$ to the power of $3$ .

$\log (8) - \log (8) + \log (100)$

Step 1.8

Logarithm base $10$ of $100$ is $2$ .

$\log (8) - \log (8) + 2$

Step 2

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

$\log (\frac{8}{8}) + 2$

Step 3

Simplify each term.

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Step 3.1

Divide $8$ by $8$ .

$\log (1) + 2$

Step 3.2

Logarithm base $10$ of $1$ is $0$ .

$0 + 2$

Step 4

Add $0$ and $2$ .

$2$