Precalculus Examples

$\ln ({(\frac{8 x}{16 x})}^{\frac{1}{2}}) + \ln (4 x^{2})$

Step 1

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

$\ln ({(\frac{8 x}{16 x})}^{\frac{1}{2}} (4 x^{2}))$

Step 2

Rewrite using the commutative property of multiplication.

$\ln (4 {(\frac{8 x}{16 x})}^{\frac{1}{2}} x^{2})$

Step 3

Cancel the common factor of $8$ and $16$ .

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Step 3.1

Factor $8$ out of $8 x$ .

$\ln (4 {(\frac{8 (x)}{16 x})}^{\frac{1}{2}} x^{2})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $8$ out of $16 x$ .

$\ln (4 {(\frac{8 (x)}{8 (2 x)})}^{\frac{1}{2}} x^{2})$

Step 3.2.2

Cancel the common factor.

$\ln (4 {(\frac{8 x}{8 (2 x)})}^{\frac{1}{2}} x^{2})$

Step 3.2.3

Rewrite the expression.

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

Step 4

Cancel the common factor of $x$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Simplify the expression.

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Step 5.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 5.2

One to any power is one.

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

Step 6

Combine $4$ and $\frac{1}{2^{\frac{1}{2}}}$ .

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

Step 7

Combine $\frac{4}{2^{\frac{1}{2}}}$ and $x^{2}$ .

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