Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $2 x$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.4

Apply the product rule to $4 x^{2}$ .

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

Step 1.5

Multiply the exponents in ${(x^{2})}^{\frac{2}{3}}$ .

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Step 1.5.1

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

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

Step 1.5.2

Multiply $2 (\frac{2}{3})$ .

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Step 1.5.2.1

Combine $2$ and $\frac{2}{3}$ .

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

Step 1.5.2.2

Multiply $2$ by $2$ .

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

Step 1.6

Apply the product rule to $4 x$ .

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

Step 1.7

Raise $4$ to the power of $2$ .

$- \log_{3} (4^{\frac{2}{3}} x^{\frac{4}{3}}) + \frac{1}{2} \cdot \log_{3} (16 x^{2})$

Step 1.8

Simplify $\frac{1}{2} \log_{3} (16 x^{2})$ by moving $\frac{1}{2}$ inside the logarithm.

$- \log_{3} (4^{\frac{2}{3}} x^{\frac{4}{3}}) + \log_{3} ({(16 x^{2})}^{\frac{1}{2}})$

Step 1.9

Apply the product rule to $16 x^{2}$ .

$- \log_{3} (4^{\frac{2}{3}} x^{\frac{4}{3}}) + \log_{3} (16^{\frac{1}{2}} {(x^{2})}^{\frac{1}{2}})$

Step 1.10

Rewrite $16$ as $4^{2}$ .

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

Step 1.11

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

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

Step 1.12

Cancel the common factor of $2$ .

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Step 1.12.1

Cancel the common factor.

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

Step 1.12.2

Rewrite the expression.

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

Step 1.13

Evaluate the exponent.

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

Step 1.14

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

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Step 1.14.1

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

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

Step 1.14.2

Cancel the common factor of $2$ .

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Step 1.14.2.1

Cancel the common factor.

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

Step 1.14.2.2

Rewrite the expression.

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

Step 1.15

Simplify.

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

Step 2

Simplify each term.

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Step 2.1

Rewrite $\log_{3} (4^{\frac{2}{3}} x^{\frac{4}{3}})$ using the change of base formula.

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Step 2.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$- \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + \log_{3} (4 x)$

Step 2.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

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

Step 2.2

Rewrite $\log_{3} (4 x)$ using the change of base formula.

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Step 2.2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$- \frac{\log (4^{\frac{2}{3}} x^{\frac{4}{3}})}{\log (3)} + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

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