Precalculus Examples

$4 \ln (7 x) - 2 \ln (x + 1) - 5 \ln (3) + \frac{2}{3} \cdot \ln (5) - \ln (x + 6) + \frac{1}{2} \cdot \ln (2 x - 3) + \frac{3}{8} \cdot \ln (x^{2} - 3 x + 11)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $4 \ln (7 x)$ by moving $4$ inside the logarithm.

$\ln ({(7 x)}^{4}) - 2 \ln (x + 1) - 5 \ln (3) + \frac{2}{3} \cdot \ln (5) - \ln (x + 6) + \frac{1}{2} \cdot \ln (2 x - 3) + \frac{3}{8} \cdot \ln (x^{2} - 3 x + 11)$

Step 1.2

Apply the product rule to $7 x$ .

$\ln (7^{4} x^{4}) - 2 \ln (x + 1) - 5 \ln (3) + \frac{2}{3} \cdot \ln (5) - \ln (x + 6) + \frac{1}{2} \cdot \ln (2 x - 3) + \frac{3}{8} \cdot \ln (x^{2} - 3 x + 11)$

Step 1.3

Raise $7$ to the power of $4$ .

$\ln (2401 x^{4}) - 2 \ln (x + 1) - 5 \ln (3) + \frac{2}{3} \cdot \ln (5) - \ln (x + 6) + \frac{1}{2} \cdot \ln (2 x - 3) + \frac{3}{8} \cdot \ln (x^{2} - 3 x + 11)$

Step 1.4

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

$\ln (2401 x^{4}) - \ln ({(x + 1)}^{2}) - 5 \ln (3) + \frac{2}{3} \cdot \ln (5) - \ln (x + 6) + \frac{1}{2} \cdot \ln (2 x - 3) + \frac{3}{8} \cdot \ln (x^{2} - 3 x + 11)$

Step 1.5

Simplify $- 5 \ln (3)$ by moving $5$ inside the logarithm.

$\ln (2401 x^{4}) - \ln ({(x + 1)}^{2}) - \ln (3^{5}) + \frac{2}{3} \cdot \ln (5) - \ln (x + 6) + \frac{1}{2} \cdot \ln (2 x - 3) + \frac{3}{8} \cdot \ln (x^{2} - 3 x + 11)$

Step 1.6

Raise $3$ to the power of $5$ .

$\ln (2401 x^{4}) - \ln ({(x + 1)}^{2}) - \ln (243) + \frac{2}{3} \cdot \ln (5) - \ln (x + 6) + \frac{1}{2} \cdot \ln (2 x - 3) + \frac{3}{8} \cdot \ln (x^{2} - 3 x + 11)$

Step 1.7

Simplify $\frac{2}{3} \ln (5)$ by moving $\frac{2}{3}$ inside the logarithm.

$\ln (2401 x^{4}) - \ln ({(x + 1)}^{2}) - \ln (243) + \ln (5^{\frac{2}{3}}) - \ln (x + 6) + \frac{1}{2} \cdot \ln (2 x - 3) + \frac{3}{8} \cdot \ln (x^{2} - 3 x + 11)$

Step 1.8

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

$\ln (2401 x^{4}) - \ln ({(x + 1)}^{2}) - \ln (243) + \ln (5^{\frac{2}{3}}) - \ln (x + 6) + \ln ({(2 x - 3)}^{\frac{1}{2}}) + \frac{3}{8} \cdot \ln (x^{2} - 3 x + 11)$

Step 1.9

Simplify $\frac{3}{8} \ln (x^{2} - 3 x + 11)$ by moving $\frac{3}{8}$ inside the logarithm.

$\ln (2401 x^{4}) - \ln ({(x + 1)}^{2}) - \ln (243) + \ln (5^{\frac{2}{3}}) - \ln (x + 6) + \ln ({(2 x - 3)}^{\frac{1}{2}}) + \ln ({(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 2

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

$\ln (\frac{2401 x^{4}}{{(x + 1)}^{2}}) - \ln (243) + \ln (5^{\frac{2}{3}}) - \ln (x + 6) + \ln ({(2 x - 3)}^{\frac{1}{2}}) + \ln ({(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 3

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

$\ln (\frac{\frac{2401 x^{4}}{{(x + 1)}^{2}}}{243}) + \ln (5^{\frac{2}{3}}) - \ln (x + 6) + \ln ({(2 x - 3)}^{\frac{1}{2}}) + \ln ({(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 4

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

$\ln (\frac{\frac{2401 x^{4}}{{(x + 1)}^{2}}}{243} \cdot 5^{\frac{2}{3}}) - \ln (x + 6) + \ln ({(2 x - 3)}^{\frac{1}{2}}) + \ln ({(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 5

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

$\ln (\frac{\frac{\frac{2401 x^{4}}{{(x + 1)}^{2}}}{243} \cdot 5^{\frac{2}{3}}}{x + 6}) + \ln ({(2 x - 3)}^{\frac{1}{2}}) + \ln ({(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 6

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

$\ln (\frac{\frac{\frac{2401 x^{4}}{{(x + 1)}^{2}}}{243} \cdot 5^{\frac{2}{3}}}{x + 6} {(2 x - 3)}^{\frac{1}{2}}) + \ln ({(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 7

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

$\ln (\frac{\frac{\frac{2401 x^{4}}{{(x + 1)}^{2}}}{243} \cdot 5^{\frac{2}{3}}}{x + 6} {(2 x - 3)}^{\frac{1}{2}} {(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 8

Combine fractions.

Tap for more steps...

Step 8.1

Combine $\frac{\frac{2401 x^{4}}{{(x + 1)}^{2}}}{243}$ and $5^{\frac{2}{3}}$ .

$\ln (\frac{\frac{\frac{2401 x^{4}}{{(x + 1)}^{2}} \cdot 5^{\frac{2}{3}}}{243}}{x + 6} {(2 x - 3)}^{\frac{1}{2}} {(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 8.2

Combine $\frac{2401 x^{4}}{{(x + 1)}^{2}}$ and $5^{\frac{2}{3}}$ .

$\ln (\frac{\frac{\frac{2401 x^{4} \cdot 5^{\frac{2}{3}}}{{(x + 1)}^{2}}}{243}}{x + 6} {(2 x - 3)}^{\frac{1}{2}} {(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 9

Simplify the numerator.

Tap for more steps...

Step 9.1

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{\frac{2401 x^{4} \cdot 5^{\frac{2}{3}}}{{(x + 1)}^{2}} \cdot \frac{1}{243}}{x + 6} {(2 x - 3)}^{\frac{1}{2}} {(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 9.2

Combine.

$\ln (\frac{\frac{2401 x^{4} \cdot 5^{\frac{2}{3}} \cdot 1}{{(x + 1)}^{2} \cdot 243}}{x + 6} {(2 x - 3)}^{\frac{1}{2}} {(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 9.3

Multiply $2401$ by $1$ .

$\ln (\frac{\frac{2401 x^{4} \cdot 5^{\frac{2}{3}}}{{(x + 1)}^{2} \cdot 243}}{x + 6} {(2 x - 3)}^{\frac{1}{2}} {(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 9.4

Move $243$ to the left of ${(x + 1)}^{2}$ .

$\ln (\frac{\frac{2401 x^{4} \cdot 5^{\frac{2}{3}}}{243 {(x + 1)}^{2}}}{x + 6} {(2 x - 3)}^{\frac{1}{2}} {(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 10

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{2401 x^{4} \cdot 5^{\frac{2}{3}}}{243 {(x + 1)}^{2}} \cdot \frac{1}{x + 6} {(2 x - 3)}^{\frac{1}{2}} {(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 11

Multiply $\frac{2401 x^{4} \cdot 5^{\frac{2}{3}}}{243 {(x + 1)}^{2}}$ by $\frac{1}{x + 6}$ .

$\ln (\frac{2401 x^{4} \cdot 5^{\frac{2}{3}}}{243 {(x + 1)}^{2} (x + 6)} {(2 x - 3)}^{\frac{1}{2}} {(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 12

Combine $\frac{2401 x^{4} \cdot 5^{\frac{2}{3}}}{243 {(x + 1)}^{2} (x + 6)}$ and ${(2 x - 3)}^{\frac{1}{2}}$ .

$\ln (\frac{2401 x^{4} \cdot 5^{\frac{2}{3}} {(2 x - 3)}^{\frac{1}{2}}}{243 {(x + 1)}^{2} (x + 6)} {(x^{2} - 3 x + 11)}^{\frac{3}{8}})$

Step 13

Combine $\frac{2401 x^{4} \cdot 5^{\frac{2}{3}} {(2 x - 3)}^{\frac{1}{2}}}{243 {(x + 1)}^{2} (x + 6)}$ and ${(x^{2} - 3 x + 11)}^{\frac{3}{8}}$ .

$\ln (\frac{2401 x^{4} \cdot 5^{\frac{2}{3}} {(2 x - 3)}^{\frac{1}{2}} {(x^{2} - 3 x + 11)}^{\frac{3}{8}}}{243 {(x + 1)}^{2} (x + 6)})$