Precalculus Examples

$3 (\ln (x) - 2 \ln (x^{2} + 1)) + 2 \ln (5)$

Step 1

Simplify each term.

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Step 1.1

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

$3 (\ln (x) - \ln ({(x^{2} + 1)}^{2})) + 2 \ln (5)$

Step 1.2

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

$3 \ln (\frac{x}{{(x^{2} + 1)}^{2}}) + 2 \ln (5)$

Step 1.3

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

$\ln ({(\frac{x}{{(x^{2} + 1)}^{2}})}^{3}) + 2 \ln (5)$

Step 1.4

Apply the product rule to $\frac{x}{{(x^{2} + 1)}^{2}}$ .

$\ln (\frac{x^{3}}{{({(x^{2} + 1)}^{2})}^{3}}) + 2 \ln (5)$

Step 1.5

Multiply the exponents in ${({(x^{2} + 1)}^{2})}^{3}$ .

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Step 1.5.1

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

$\ln (\frac{x^{3}}{{(x^{2} + 1)}^{2 \cdot 3}}) + 2 \ln (5)$

Step 1.5.2

Multiply $2$ by $3$ .

$\ln (\frac{x^{3}}{{(x^{2} + 1)}^{6}}) + 2 \ln (5)$

Step 1.6

Simplify $2 \ln (5)$ by moving $2$ inside the logarithm.

$\ln (\frac{x^{3}}{{(x^{2} + 1)}^{6}}) + \ln (5^{2})$

Step 1.7

Raise $5$ to the power of $2$ .

$\ln (\frac{x^{3}}{{(x^{2} + 1)}^{6}}) + \ln (25)$

Step 2

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

$\ln (\frac{x^{3}}{{(x^{2} + 1)}^{6}} \cdot 25)$

Step 3

Combine $\frac{x^{3}}{{(x^{2} + 1)}^{6}}$ and $25$ .

$\ln (\frac{x^{3} \cdot 25}{{(x^{2} + 1)}^{6}})$

Step 4

Move $25$ to the left of $x^{3}$ .

$\ln (\frac{25 x^{3}}{{(x^{2} + 1)}^{6}})$