Precalculus Examples

$2 \ln (\sqrt{x}) + 3 \ln (2 x)$

Step 1

Simplify each term.

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Step 1.1

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

$\ln ({\sqrt{x}}^{2}) + 3 \ln (2 x)$

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

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

Step 1.2.5

Simplify.

$\ln (x) + 3 \ln (2 x)$

Step 1.3

Simplify $3 \ln (2 x)$ by moving $3$ inside the logarithm.

$\ln (x) + \ln ({(2 x)}^{3})$

Step 1.4

Apply the product rule to $2 x$ .

$\ln (x) + \ln (2^{3} x^{3})$

Step 1.5

Raise $2$ to the power of $3$ .

$\ln (x) + \ln (8 x^{3})$

Step 2

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

$\ln (x (8 x^{3}))$

Step 3

Rewrite using the commutative property of multiplication.

$\ln (8 x \cdot x^{3})$

Step 4

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 4.1

Move $x^{3}$ .

$\ln (8 (x^{3} x))$

Step 4.2

Multiply $x^{3}$ by $x$ .

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Step 4.2.1

Raise $x$ to the power of $1$ .

$\ln (8 (x^{3} x^{1}))$

Step 4.2.2

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

$\ln (8 x^{3 + 1})$

Step 4.3

Add $3$ and $1$ .

$\ln (8 x^{4})$