Precalculus Examples

$3 \log (x - 1) - 2 \log (4 x + 2) + \frac{1}{3} \cdot \log (x - 1)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (x - 1)$ by moving $3$ inside the logarithm.

$\log ({(x - 1)}^{3}) - 2 \log (4 x + 2) + \frac{1}{3} \cdot \log (x - 1)$

Step 1.2

Simplify $- 2 \log (4 x + 2)$ by moving $2$ inside the logarithm.

$\log ({(x - 1)}^{3}) - \log ({(4 x + 2)}^{2}) + \frac{1}{3} \cdot \log (x - 1)$

Step 1.3

Simplify $\frac{1}{3} \log (x - 1)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log ({(x - 1)}^{3}) - \log ({(4 x + 2)}^{2}) + \log ({(x - 1)}^{\frac{1}{3}})$

Step 2

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

$\log (\frac{{(x - 1)}^{3}}{{(4 x + 2)}^{2}}) + \log ({(x - 1)}^{\frac{1}{3}})$

Step 3

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

$\log (\frac{{(x - 1)}^{3}}{{(4 x + 2)}^{2}} {(x - 1)}^{\frac{1}{3}})$

Step 4

Simplify the denominator.

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Step 4.1

Factor $2$ out of $4 x + 2$ .

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Step 4.1.1

Factor $2$ out of $4 x$ .

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

Step 4.1.2

Factor $2$ out of $2$ .

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

Step 4.1.3

Factor $2$ out of $2 (2 x) + 2 (1)$ .

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

Step 4.2

Apply the product rule to $2 (2 x + 1)$ .

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

Step 4.3

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

$\log (\frac{{(x - 1)}^{3}}{4 {(2 x + 1)}^{2}} {(x - 1)}^{\frac{1}{3}})$

Step 5

Multiply $\frac{{(x - 1)}^{3}}{4 {(2 x + 1)}^{2}} {(x - 1)}^{\frac{1}{3}}$ .

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Step 5.1

Combine $\frac{{(x - 1)}^{3}}{4 {(2 x + 1)}^{2}}$ and ${(x - 1)}^{\frac{1}{3}}$ .

$\log (\frac{{(x - 1)}^{3} {(x - 1)}^{\frac{1}{3}}}{4 {(2 x + 1)}^{2}})$

Step 5.2

Multiply ${(x - 1)}^{3}$ by ${(x - 1)}^{\frac{1}{3}}$ by adding the exponents.

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Step 5.2.1

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

$\log (\frac{{(x - 1)}^{3 + \frac{1}{3}}}{4 {(2 x + 1)}^{2}})$

Step 5.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log (\frac{{(x - 1)}^{3 \cdot \frac{3}{3} + \frac{1}{3}}}{4 {(2 x + 1)}^{2}})$

Step 5.2.3

Combine $3$ and $\frac{3}{3}$ .

$\log (\frac{{(x - 1)}^{\frac{3 \cdot 3}{3} + \frac{1}{3}}}{4 {(2 x + 1)}^{2}})$

Step 5.2.4

Combine the numerators over the common denominator.

$\log (\frac{{(x - 1)}^{\frac{3 \cdot 3 + 1}{3}}}{4 {(2 x + 1)}^{2}})$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $3$ by $3$ .

$\log (\frac{{(x - 1)}^{\frac{9 + 1}{3}}}{4 {(2 x + 1)}^{2}})$

Step 5.2.5.2

Add $9$ and $1$ .

$\log (\frac{{(x - 1)}^{\frac{10}{3}}}{4 {(2 x + 1)}^{2}})$