Simplify log of 9x^5+5 log of 1/x

$\log (9 x^{5}) + 5 \log (\frac{1}{x})$

Simplify each term.

Simplify $5 \log (\frac{1}{x})$ by moving $5$ inside the logarithm.

$\log (9 x^{5}) + \log ({(\frac{1}{x})}^{5})$

Apply the product rule to $\frac{1}{x}$ .

$\log (9 x^{5}) + \log (\frac{1^{5}}{x^{5}})$

One to any power is one.

$\log (9 x^{5}) + \log (\frac{1}{x^{5}})$

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

$\log (9 x^{5} \frac{1}{x^{5}})$

Cancel the common factor of $x^{5}$ .

Factor $x^{5}$ out of $9 x^{5}$ .

$\log (x^{5} \cdot 9 \frac{1}{x^{5}})$

Cancel the common factor.

$\log (x^{5} \cdot 9 \frac{1}{x^{5}})$

Rewrite the expression.

$\log (9)$

The result can be shown in multiple forms.

Exact Form:

$\log (9)$

Decimal Form:

$0.95424250 \dots$