Precalculus Examples

$- 2 \log (\sqrt{y}) + \frac{5}{2} \cdot \log (9 x^{4}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1

Simplify each term.

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Step 1.1

Simplify $- 2 \log (\sqrt{y})$ by moving $2$ inside the logarithm.

$- \log ({\sqrt{y}}^{2}) + \frac{5}{2} \cdot \log (9 x^{4}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.2

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$- \log ({(y^{\frac{1}{2}})}^{2}) + \frac{5}{2} \cdot \log (9 x^{4}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.2.2

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

$- \log (y^{\frac{1}{2} \cdot 2}) + \frac{5}{2} \cdot \log (9 x^{4}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.2.3

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

$- \log (y^{\frac{2}{2}}) + \frac{5}{2} \cdot \log (9 x^{4}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

$- \log (y^{\frac{2}{2}}) + \frac{5}{2} \cdot \log (9 x^{4}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.2.4.2

Rewrite the expression.

$- \log (y^{1}) + \frac{5}{2} \cdot \log (9 x^{4}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.2.5

Simplify.

$- \log (y) + \frac{5}{2} \cdot \log (9 x^{4}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.3

Simplify $\frac{5}{2} \log (9 x^{4})$ by moving $\frac{5}{2}$ inside the logarithm.

$- \log (y) + \log ({(9 x^{4})}^{\frac{5}{2}}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.4

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

$- \log (y) + \log (9^{\frac{5}{2}} {(x^{4})}^{\frac{5}{2}}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.5

Rewrite $9$ as $3^{2}$ .

$- \log (y) + \log ({(3^{2})}^{\frac{5}{2}} {(x^{4})}^{\frac{5}{2}}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.6

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

$- \log (y) + \log (3^{2 (\frac{5}{2})} {(x^{4})}^{\frac{5}{2}}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.7

Cancel the common factor of $2$ .

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Step 1.7.1

Cancel the common factor.

$- \log (y) + \log (3^{2 (\frac{5}{2})} {(x^{4})}^{\frac{5}{2}}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.7.2

Rewrite the expression.

$- \log (y) + \log (3^{5} {(x^{4})}^{\frac{5}{2}}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.8

Raise $3$ to the power of $5$ .

$- \log (y) + \log (243 {(x^{4})}^{\frac{5}{2}}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.9

Multiply the exponents in ${(x^{4})}^{\frac{5}{2}}$ .

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Step 1.9.1

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

$- \log (y) + \log (243 x^{4 (\frac{5}{2})}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.9.2

Cancel the common factor of $2$ .

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Step 1.9.2.1

Factor $2$ out of $4$ .

$- \log (y) + \log (243 x^{2 (2) \frac{5}{2}}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.9.2.2

Cancel the common factor.

$- \log (y) + \log (243 x^{2 \cdot 2 \frac{5}{2}}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.9.2.3

Rewrite the expression.

$- \log (y) + \log (243 x^{2 \cdot 5}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.9.3

Multiply $2$ by $5$ .

$- \log (y) + \log (243 x^{10}) + \frac{1}{2} \cdot \log (2 \sqrt{x})$

Step 1.10

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

$- \log (y) + \log (243 x^{10}) + \log ({(2 \sqrt{x})}^{\frac{1}{2}})$

Step 1.11

Apply the product rule to $2 \sqrt{x}$ .

$- \log (y) + \log (243 x^{10}) + \log (2^{\frac{1}{2}} {\sqrt{x}}^{\frac{1}{2}})$

Step 2

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

$- \log (y) + \log (243 x^{10} (2^{\frac{1}{2}} {\sqrt{x}}^{\frac{1}{2}}))$

Step 3

Rewrite using the commutative property of multiplication.

$- \log (y) + \log (243 \cdot 2^{\frac{1}{2}} x^{10} {\sqrt{x}}^{\frac{1}{2}})$

Step 4

Reorder factors in $- \log (y) + \log (243 \cdot 2^{\frac{1}{2}} x^{10} {\sqrt{x}}^{\frac{1}{2}})$ .

$- \log (y) + \log (243 x^{10} \cdot 2^{\frac{1}{2}} {\sqrt{x}}^{\frac{1}{2}})$