Algebra Examples

$\frac{1}{2} (\log_{5} (x^{6}) + \log_{5} (x^{3}))$

Step 1

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

$\frac{1}{2} \cdot \log_{5} (x^{6} x^{3})$

Step 2

Multiply $x^{6}$ by $x^{3}$ by adding the exponents.

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Step 2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} \cdot \log_{5} (x^{6 + 3})$

Step 2.2

Add $6$ and $3$ .

$\frac{1}{2} \cdot \log_{5} (x^{9})$

Step 3

Simplify $\frac{1}{2} \log_{5} (x^{9})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{5} ({(x^{9})}^{\frac{1}{2}})$

Step 4

Multiply the exponents in ${(x^{9})}^{\frac{1}{2}}$ .

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Step 4.1

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

$\log_{5} (x^{9 (\frac{1}{2})})$

Step 4.2

Combine $9$ and $\frac{1}{2}$ .

$\log_{5} (x^{\frac{9}{2}})$