Algebra Examples

\frac{1}{2} {log}_{5} (15) - {log}_{5} (\sqrt{75})

$\frac{1}{2} \log_{5} (15) - \log_{5} (\sqrt{75})$

Step 1

Simplify each term.

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Step 1.1

Simplify

$\frac{1}{2} \log_{5} (15)$ by moving

$\frac{1}{2}$ inside the logarithm.

$\log_{5} (15^{\frac{1}{2}}) - \log_{5} (\sqrt{75})$

Step 1.2

Rewrite

$75$ as

$5^{2} \cdot 3$ .

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Step 1.2.1

Factor

$25$ out of

$75$ .

$\log_{5} (15^{\frac{1}{2}}) - \log_{5} (\sqrt{25 (3)})$

Step 1.2.2

Rewrite

$25$ as

$5^{2}$ .

$\log_{5} (15^{\frac{1}{2}}) - \log_{5} (\sqrt{5^{2} \cdot 3})$

Step 1.3

Pull terms out from under the radical.

$\log_{5} (15^{\frac{1}{2}}) - \log_{5} (5 \sqrt{3})$

Step 2

Use the quotient property of logarithms,

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{15^{\frac{1}{2}}}{5 \sqrt{3}})$

Step 3

Multiply

$\frac{15^{\frac{1}{2}}}{5 \sqrt{3}}$ by

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$\log_{5} (\frac{15^{\frac{1}{2}}}{5 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply

$\frac{15^{\frac{1}{2}}}{5 \sqrt{3}}$ by

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 \sqrt{3} \sqrt{3}})$

Step 4.2

Move

$\sqrt{3}$ .

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 (\sqrt{3} \sqrt{3})})$

Step 4.3

Raise

$\sqrt{3}$ to the power of

$1$ .

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 ({\sqrt{3}}^{1} \sqrt{3})})$

Step 4.4

Raise

$\sqrt{3}$ to the power of

$1$ .

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})})$

Step 4.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 {\sqrt{3}}^{1 + 1}})$

Step 4.6

Add

$1$ and

$1$ .

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 {\sqrt{3}}^{2}})$

Step 4.7

Rewrite

${\sqrt{3}}^{2}$ as

$3$ .

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Step 4.7.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{3}$ as

$3^{\frac{1}{2}}$ .

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 {(3^{\frac{1}{2}})}^{2}})$

Step 4.7.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 \cdot 3^{\frac{1}{2} \cdot 2}})$

Step 4.7.3

Combine

$\frac{1}{2}$ and

$2$ .

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 \cdot 3^{\frac{2}{2}}})$

Step 4.7.4

Cancel the common factor of

$2$ .

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Step 4.7.4.1

Cancel the common factor.

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 \cdot 3^{\frac{2}{2}}})$

Step 4.7.4.2

Rewrite the expression.

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 \cdot 3^{1}})$

Step 4.7.5

Evaluate the exponent.

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{5 \cdot 3})$

Step 5

Multiply.

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Step 5.1

Multiply

$5$ by

$3$ .

$\log_{5} (\frac{15^{\frac{1}{2}} \sqrt{3}}{15})$

Step 5.2

Move

$15^{\frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{n} = \frac{1}{b^{- n}}$ .

$\log_{5} (\frac{\sqrt{3}}{15 \cdot 15^{- \frac{1}{2}}})$

Step 6

Multiply

$15$ by

$15^{- \frac{1}{2}}$ by adding the exponents.

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Step 6.1

Multiply

$15$ by

$15^{- \frac{1}{2}}$ .

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Step 6.1.1

Raise

$15$ to the power of

$1$ .

$\log_{5} (\frac{\sqrt{3}}{15^{1} \cdot 15^{- \frac{1}{2}}})$

Step 6.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{5} (\frac{\sqrt{3}}{15^{1 - \frac{1}{2}}})$

Step 6.2

Write

$1$ as a fraction with a common denominator.

$\log_{5} (\frac{\sqrt{3}}{15^{\frac{2}{2} - \frac{1}{2}}})$

Step 6.3

Combine the numerators over the common denominator.

$\log_{5} (\frac{\sqrt{3}}{15^{\frac{2 - 1}{2}}})$

Step 6.4

Subtract

$1$ from

$2$ .

$\log_{5} (\frac{\sqrt{3}}{15^{\frac{1}{2}}})$

Step 7

Rewrite

$\log_{5} (\frac{\sqrt{3}}{15^{\frac{1}{2}}})$ using the change of base formula.

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Step 7.1

The change of base rule can be used if

$a$ and

$b$ are greater than

$0$ and not equal to

$1$ , and

$x$ is greater than

$0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 7.2

Substitute in values for the variables in the change of base formula, using

$b = 10$ .

$\frac{\log (\frac{\sqrt{3}}{15^{\frac{1}{2}}})}{\log (5)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{\sqrt{3}}{15^{\frac{1}{2}}})}{\log (5)}$

Decimal Form:

$- 0.5$