Precalculus Examples

$\frac{1}{2} \cdot \log_{9} (9 x^{4}) - 6 \log_{b} (y)$

Step 1

Simplify $\frac{1}{2} \log_{9} (9 x^{4})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{9} ({(9 x^{4})}^{\frac{1}{2}}) - 6 \log_{b} (y)$

Step 2

Apply the product rule to $9 x^{4}$ .

$\log_{9} (9^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}) - 6 \log_{b} (y)$

Step 3

Rewrite $\log_{9} (9^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}})$ as $\log_{9} (9^{\frac{1}{2}}) + \log_{9} ({(x^{4})}^{\frac{1}{2}})$ .

$\log_{9} (9^{\frac{1}{2}}) + \log_{9} ({(x^{4})}^{\frac{1}{2}}) - 6 \log_{b} (y)$

Step 4

Use logarithm rules to move $\frac{1}{2}$ out of the exponent.

$\frac{1}{2} \log_{9} (9) + \log_{9} ({(x^{4})}^{\frac{1}{2}}) - 6 \log_{b} (y)$

Step 5

Logarithm base $9$ of $9$ is $1$ .

$\frac{1}{2} \cdot 1 + \log_{9} ({(x^{4})}^{\frac{1}{2}}) - 6 \log_{b} (y)$

Step 6

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} + \log_{9} ({(x^{4})}^{\frac{1}{2}}) - 6 \log_{b} (y)$

Step 7

Multiply the exponents in ${(x^{4})}^{\frac{1}{2}}$ .

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Step 7.1

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

$\frac{1}{2} + \log_{9} (x^{4 (\frac{1}{2})}) - 6 \log_{b} (y)$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Factor $2$ out of $4$ .

$\frac{1}{2} + \log_{9} (x^{2 (2) \frac{1}{2}}) - 6 \log_{b} (y)$

Step 7.2.2

Cancel the common factor.

$\frac{1}{2} + \log_{9} (x^{2 \cdot 2 \frac{1}{2}}) - 6 \log_{b} (y)$

Step 7.2.3

Rewrite the expression.

$\frac{1}{2} + \log_{9} (x^{2}) - 6 \log_{b} (y)$

Step 8

Simplify $- 6 \log_{b} (y)$ by moving $6$ inside the logarithm.

$\frac{1}{2} + \log_{9} (x^{2}) - \log_{b} (y^{6})$