Precalculus Examples

$- \frac{3}{2} \cdot (\log (x) + 2 \log (2))$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log (2)$ by moving $2$ inside the logarithm.

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

Step 1.2

Raise $2$ to the power of $2$ .

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

Step 2

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

$- \frac{3}{2} \cdot \log (x \cdot 4)$

Step 3

Move $4$ to the left of $x$ .

$- \frac{3}{2} \cdot \log (4 \cdot x)$

Step 4

Multiply $- \frac{3}{2} \log (4 x)$ .

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Step 4.1

Reorder $\log (4 x)$ and $\frac{3}{2}$ .

$- (\frac{3}{2} \log (4 x))$

Step 4.2

Simplify $\frac{3}{2} \log (4 x)$ by moving $\frac{3}{2}$ inside the logarithm.

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

Step 5

Simplify the expression.

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Step 5.1

Apply the product rule to $4 x$ .

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

Step 5.2

Rewrite $4$ as $2^{2}$ .

$- \log ({(2^{2})}^{\frac{3}{2}} x^{\frac{3}{2}})$

Step 5.3

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

$- \log (2^{2 (\frac{3}{2})} x^{\frac{3}{2}})$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Cancel the common factor.

$- \log (2^{2 (\frac{3}{2})} x^{\frac{3}{2}})$

Step 6.2

Rewrite the expression.

$- \log (2^{3} x^{\frac{3}{2}})$

Step 7

Raise $2$ to the power of $3$ .

$- \log (8 x^{\frac{3}{2}})$