Precalculus Examples

$4 \log_{5} (\sqrt{3 x - 5}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 5}$ as ${(3 x - 5)}^{\frac{1}{2}}$ .

$4 \log_{5} ({(3 x - 5)}^{\frac{1}{2}}) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 2

Expand $\log_{5} ({(3 x - 5)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$4 (\frac{1}{2} \log_{5} (3 x - 5)) - \log_{5} (\frac{2}{x}) + \log_{5} (2)$

Step 3

Rewrite $\log_{5} (\frac{2}{x})$ as $\log_{5} (2) - \log_{5} (x)$ .

$4 (\frac{1}{2} \log_{5} (3 x - 5)) - (\log_{5} (2) - \log_{5} (x)) + \log_{5} (2)$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Combine $\frac{1}{2}$ and $\log_{5} (3 x - 5)$ .

$4 \frac{\log_{5} (3 x - 5)}{2} - (\log_{5} (2) - \log_{5} (x)) + \log_{5} (2)$

Step 4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.1

Factor $2$ out of $4$ .

$2 (2) \frac{\log_{5} (3 x - 5)}{2} - (\log_{5} (2) - \log_{5} (x)) + \log_{5} (2)$

Step 4.2.2

Cancel the common factor.

$2 \cdot 2 \frac{\log_{5} (3 x - 5)}{2} - (\log_{5} (2) - \log_{5} (x)) + \log_{5} (2)$

Step 4.2.3

Rewrite the expression.

$2 \log_{5} (3 x - 5) - (\log_{5} (2) - \log_{5} (x)) + \log_{5} (2)$

Step 4.3

Apply the distributive property.

$2 \log_{5} (3 x - 5) - \log_{5} (2) - - \log_{5} (x) + \log_{5} (2)$

Step 4.4

Multiply $- - \log_{5} (x)$ .

Tap for more steps...

Step 4.4.1

Multiply $- 1$ by $- 1$ .

$2 \log_{5} (3 x - 5) - \log_{5} (2) + 1 \log_{5} (x) + \log_{5} (2)$

Step 4.4.2

Multiply $\log_{5} (x)$ by $1$ .

$2 \log_{5} (3 x - 5) - \log_{5} (2) + \log_{5} (x) + \log_{5} (2)$

Step 5

Add $- \log_{5} (2)$ and $\log_{5} (2)$ .

$2 \log_{5} (3 x - 5) + 0 + \log_{5} (x)$

Step 6

Add $2 \log_{5} (3 x - 5)$ and $0$ .

$2 \log_{5} (3 x - 5) + \log_{5} (x)$