Precalculus Examples

$\log_{7} (\frac{e^{2 x} {(x + 1)}^{7}}{(x^{2} + 4) {(x - 3)}^{\frac{1}{3}}})$

Step 1

Rewrite $\log_{7} (\frac{e^{2 x} {(x + 1)}^{7}}{(x^{2} + 4) {(x - 3)}^{\frac{1}{3}}})$ as $\log_{7} (e^{2 x} {(x + 1)}^{7}) - \log_{7} ((x^{2} + 4) {(x - 3)}^{\frac{1}{3}})$ .

$\log_{7} (e^{2 x} {(x + 1)}^{7}) - \log_{7} ((x^{2} + 4) {(x - 3)}^{\frac{1}{3}})$

Step 2

Rewrite $\log_{7} (e^{2 x} {(x + 1)}^{7})$ as $\log_{7} (e^{2 x}) + \log_{7} ({(x + 1)}^{7})$ .

$\log_{7} (e^{2 x}) + \log_{7} ({(x + 1)}^{7}) - \log_{7} ((x^{2} + 4) {(x - 3)}^{\frac{1}{3}})$

Step 3

Expand $\log_{7} (e^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \log_{7} (e) + \log_{7} ({(x + 1)}^{7}) - \log_{7} ((x^{2} + 4) {(x - 3)}^{\frac{1}{3}})$

Step 4

Expand $\log_{7} ({(x + 1)}^{7})$ by moving $7$ outside the logarithm.

$2 x \log_{7} (e) + 7 \log_{7} (x + 1) - \log_{7} ((x^{2} + 4) {(x - 3)}^{\frac{1}{3}})$

Step 5

Rewrite $\log_{7} ((x^{2} + 4) {(x - 3)}^{\frac{1}{3}})$ as $\log_{7} (x^{2} + 4) + \log_{7} ({(x - 3)}^{\frac{1}{3}})$ .

$2 x \log_{7} (e) + 7 \log_{7} (x + 1) - (\log_{7} (x^{2} + 4) + \log_{7} ({(x - 3)}^{\frac{1}{3}}))$

Step 6

Expand $\log_{7} ({(x - 3)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$2 x \log_{7} (e) + 7 \log_{7} (x + 1) - (\log_{7} (x^{2} + 4) + \frac{1}{3} \log_{7} (x - 3))$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{1}{3}$ and $\log_{7} (x - 3)$ .

$2 x \log_{7} (e) + 7 \log_{7} (x + 1) - (\log_{7} (x^{2} + 4) + \frac{\log_{7} (x - 3)}{3})$

Step 7.2

To write $\log_{7} (x^{2} + 4)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 x \log_{7} (e) + 7 \log_{7} (x + 1) - (\log_{7} (x^{2} + 4) \cdot \frac{3}{3} + \frac{\log_{7} (x - 3)}{3})$

Step 7.3

Combine $\log_{7} (x^{2} + 4)$ and $\frac{3}{3}$ .

$2 x \log_{7} (e) + 7 \log_{7} (x + 1) - (\frac{\log_{7} (x^{2} + 4) \cdot 3}{3} + \frac{\log_{7} (x - 3)}{3})$

Step 7.4

Combine the numerators over the common denominator.

$2 x \log_{7} (e) + 7 \log_{7} (x + 1) - \frac{\log_{7} (x^{2} + 4) \cdot 3 + \log_{7} (x - 3)}{3}$

Step 7.5

Move $3$ to the left of $\log_{7} (x^{2} + 4)$ .

$2 x \log_{7} (e) + 7 \log_{7} (x + 1) - \frac{3 \log_{7} (x^{2} + 4) + \log_{7} (x - 3)}{3}$

Step 8

To write $7 \log_{7} (x + 1)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 x \log_{7} (e) + 7 \log_{7} (x + 1) \cdot \frac{3}{3} - \frac{3 \log_{7} (x^{2} + 4) + \log_{7} (x - 3)}{3}$

Step 9

Combine $7 \log_{7} (x + 1)$ and $\frac{3}{3}$ .

$2 x \log_{7} (e) + \frac{7 \log_{7} (x + 1) \cdot 3}{3} - \frac{3 \log_{7} (x^{2} + 4) + \log_{7} (x - 3)}{3}$

Step 10

Combine the numerators over the common denominator.

$2 x \log_{7} (e) + \frac{7 \log_{7} (x + 1) \cdot 3 - (3 \log_{7} (x^{2} + 4) + \log_{7} (x - 3))}{3}$

Step 11

Simplify the numerator.

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Step 11.1

Multiply $3$ by $7$ .

$2 x \log_{7} (e) + \frac{21 \log_{7} (x + 1) - (3 \log_{7} (x^{2} + 4) + \log_{7} (x - 3))}{3}$

Step 11.2

Apply the distributive property.

$2 x \log_{7} (e) + \frac{21 \log_{7} (x + 1) - (3 \log_{7} (x^{2} + 4)) - \log_{7} (x - 3)}{3}$

Step 11.3

Multiply $3$ by $- 1$ .

$2 x \log_{7} (e) + \frac{21 \log_{7} (x + 1) - 3 \log_{7} (x^{2} + 4) - \log_{7} (x - 3)}{3}$