Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.2

Multiply the exponents in ${(x^{2})}^{\frac{1}{2}}$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

Cancel the common factor of $2$ .

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Step 1.1.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2

Rewrite the expression.

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

Step 1.1.3

Simplify.

$\ln (2 x - 1) + 3 \ln (x - \ln (x)) + 2 \ln (x - 1)$

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

Rewrite $- 1 {(x - \ln (x))}^{3}$ as $- {(x - \ln (x))}^{3}$ .

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

Step 6

Apply the distributive property.

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