Expand the Logarithmic Expression log base 13 of (3x^4)/( cube root of 7x-3)

$\log_{13} (\frac{3 x^{4}}{\sqrt[3]{7 x - 3}})$

Step 1

Rewrite $\log_{13} (\frac{3 x^{4}}{\sqrt[3]{7 x - 3}})$ as $\log_{13} (3 x^{4}) - \log_{13} (\sqrt[3]{7 x - 3})$ .

$\log_{13} (3 x^{4}) - \log_{13} (\sqrt[3]{7 x - 3})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7 x - 3}$ as ${(7 x - 3)}^{\frac{1}{3}}$ .

$\log_{13} (3 x^{4}) - \log_{13} ({(7 x - 3)}^{\frac{1}{3}})$

Step 3

Expand $\log_{13} ({(7 x - 3)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log_{13} (3 x^{4}) - (\frac{1}{3} \log_{13} (7 x - 3))$

Step 4

Combine $\frac{1}{3}$ and $\log_{13} (7 x - 3)$ .

$\log_{13} (3 x^{4}) - \frac{\log_{13} (7 x - 3)}{3}$

Step 5

Rewrite $\log_{13} (3 x^{4})$ as $\log_{13} (3) + \log_{13} (x^{4})$ .

$\log_{13} (3) + \log_{13} (x^{4}) - \frac{\log_{13} (7 x - 3)}{3}$

Step 6

Expand $\log_{13} (x^{4})$ by moving $4$ outside the logarithm.

$\log_{13} (3) + 4 \log_{13} (x) - \frac{\log_{13} (7 x - 3)}{3}$

Step 7

To write $\log_{13} (3)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 \log_{13} (x) + \log_{13} (3) \cdot \frac{3}{3} - \frac{\log_{13} (7 x - 3)}{3}$

Step 8

Combine $\log_{13} (3)$ and $\frac{3}{3}$ .

$4 \log_{13} (x) + \frac{\log_{13} (3) \cdot 3}{3} - \frac{\log_{13} (7 x - 3)}{3}$

Step 9

Combine the numerators over the common denominator.

$4 \log_{13} (x) + \frac{\log_{13} (3) \cdot 3 - \log_{13} (7 x - 3)}{3}$

Step 10

Move $3$ to the left of $\log_{13} (3)$ .

$4 \log_{13} (x) + \frac{3 \log_{13} (3) - \log_{13} (7 x - 3)}{3}$