Precalculus Examples

$\frac{1}{2} \cdot (\ln (x) + \ln (4) - \ln (9))$

Step 1

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

$\frac{1}{2} \cdot (\ln (x \cdot 4) - \ln (9))$

Step 2

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

$\frac{1}{2} \cdot \ln (\frac{x \cdot 4}{9})$

Step 3

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

$\frac{1}{2} \cdot \ln (\frac{4 \cdot x}{9})$

Step 4

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

$\ln ({(\frac{4 x}{9})}^{\frac{1}{2}})$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

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

$\ln (\frac{{(4 x)}^{\frac{1}{2}}}{9^{\frac{1}{2}}})$

Step 5.2

Apply the product rule to $4 x$ .

$\ln (\frac{4^{\frac{1}{2}} x^{\frac{1}{2}}}{9^{\frac{1}{2}}})$

Step 6

Simplify the numerator.

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Step 6.1

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

$\ln (\frac{{(2^{2})}^{\frac{1}{2}} x^{\frac{1}{2}}}{9^{\frac{1}{2}}})$

Step 6.2

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

$\ln (\frac{2^{2 (\frac{1}{2})} x^{\frac{1}{2}}}{9^{\frac{1}{2}}})$

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Cancel the common factor.

$\ln (\frac{2^{2 (\frac{1}{2})} x^{\frac{1}{2}}}{9^{\frac{1}{2}}})$

Step 6.3.2

Rewrite the expression.

$\ln (\frac{2^{1} x^{\frac{1}{2}}}{9^{\frac{1}{2}}})$

Step 6.4

Evaluate the exponent.

$\ln (\frac{2 x^{\frac{1}{2}}}{9^{\frac{1}{2}}})$

Step 7

Simplify the denominator.

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Step 7.1

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

$\ln (\frac{2 x^{\frac{1}{2}}}{{(3^{2})}^{\frac{1}{2}}})$

Step 7.2

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

$\ln (\frac{2 x^{\frac{1}{2}}}{3^{2 (\frac{1}{2})}})$

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Cancel the common factor.

$\ln (\frac{2 x^{\frac{1}{2}}}{3^{2 (\frac{1}{2})}})$

Step 7.3.2

Rewrite the expression.

$\ln (\frac{2 x^{\frac{1}{2}}}{3^{1}})$

Step 7.4

Evaluate the exponent.

$\ln (\frac{2 x^{\frac{1}{2}}}{3})$