Precalculus Examples

$(\frac{1}{2}) (2 \log (3) + 4 \log (5) - \log (2))$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{1}{2} (\log (3^{2}) + 4 \log (5) - \log (2))$

Step 1.2

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

$\frac{1}{2} (\log (9) + 4 \log (5) - \log (2))$

Step 1.3

Simplify $4 \log (5)$ by moving $4$ inside the logarithm.

$\frac{1}{2} (\log (9) + \log (5^{4}) - \log (2))$

Step 1.4

Raise $5$ to the power of $4$ .

$\frac{1}{2} (\log (9) + \log (625) - \log (2))$

Step 2

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

$\frac{1}{2} (\log (9 \cdot 625) - \log (2))$

Step 3

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

$\frac{1}{2} \log (\frac{9 \cdot 625}{2})$

Step 4

Multiply $9$ by $625$ .

$\frac{1}{2} \log (\frac{5625}{2})$

Step 5

Simplify $\frac{1}{2} \log (\frac{5625}{2})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log ({(\frac{5625}{2})}^{\frac{1}{2}})$

Step 6

Apply the product rule to $\frac{5625}{2}$ .

$\log (\frac{5625^{\frac{1}{2}}}{2^{\frac{1}{2}}})$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $5625$ as $75^{2}$ .

$\log (\frac{{(75^{2})}^{\frac{1}{2}}}{2^{\frac{1}{2}}})$

Step 7.2

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

$\log (\frac{75^{2 (\frac{1}{2})}}{2^{\frac{1}{2}}})$

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Cancel the common factor.

$\log (\frac{75^{2 (\frac{1}{2})}}{2^{\frac{1}{2}}})$

Step 7.3.2

Rewrite the expression.

$\log (\frac{75^{1}}{2^{\frac{1}{2}}})$

Step 7.4

Evaluate the exponent.

$\log (\frac{75}{2^{\frac{1}{2}}})$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\log (\frac{75}{2^{\frac{1}{2}}})$

Decimal Form:

$1.72454626 \dots$