Algebra Examples

$(\log (9) + \frac{1}{2} \log (x) + \log (x^{3} + 4)) - \log (6)$

Step 1

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

$\frac{1}{2} \cdot \log (x) + \log (9 (x^{3} + 4)) - \log (6)$

Step 2

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

$\frac{1}{2} \cdot \log (x) + \log (\frac{9 (x^{3} + 4)}{6})$

Step 3

Simplify each term.

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Step 3.1

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

$\log (x^{\frac{1}{2}}) + \log (\frac{9 (x^{3} + 4)}{6})$

Step 3.2

Cancel the common factor of $9$ and $6$ .

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Step 3.2.1

Factor $3$ out of $9 (x^{3} + 4)$ .

$\log (x^{\frac{1}{2}}) + \log (\frac{3 (3 (x^{3} + 4))}{6})$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $3$ out of $6$ .

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

Step 3.2.2.2

Cancel the common factor.

$\log (x^{\frac{1}{2}}) + \log (\frac{3 (3 (x^{3} + 4))}{3 \cdot 2})$

Step 3.2.2.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Combine $x^{\frac{1}{2}}$ and $\frac{3 (x^{3} + 4)}{2}$ .

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

Step 6

Move $3$ to the left of $x^{\frac{1}{2}}$ .

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