Algebra Examples

$\frac{1}{2} \ln (x) + 2 \ln (x)$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Multiply $x^{\frac{1}{2}}$ by $x^{2}$ by adding the exponents.

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Step 3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $2$ .

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

Step 3.5.2

Add $1$ and $4$ .

$\ln (x^{\frac{5}{2}})$