Algebra Examples

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

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:

$- 0.5$