What is log_{10}(9x^5)+5log_{10}(1/x) ?

The solution to log_{10}(9x^5)+5log_{10}(1/x) is 2log_{10}(3)

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simplify log_{10}(9x^5)+5log_{10}(1/x)

simplify

lo g_{10} (9 x^{5}) + 5 lo g_{10} (\frac{1}{x})

2 lo g_{10} (3)

Decimal Notation

0.95424 \dots

Solution steps

lo g_{10} (9 x^{5}) + 5 lo g_{10} (\frac{1}{x})

Apply log rule:

b \cdot lo g_{a} (x) = lo g_{a} (x^{b})

5 lo g_{10} (\frac{1}{x}) = lo g_{10} ((\frac{1}{x})^{5})

= lo g_{10} (9 x^{5}) + lo g_{10} ((\frac{1}{x})^{5})

Apply log rule:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

lo g_{10} (9 x^{5}) + lo g_{10} ((\frac{1}{x})^{5}) = lo g_{10} (9 x^{5} (\frac{1}{x})^{5})

= lo g_{10} (9 x^{5} (\frac{1}{x})^{5})

Simplify

9 x^{5} (\frac{1}{x})^{5} : 9

= lo g_{10} (9)

Factor the number:

9 = 3^{2}

= lo g_{10} (3^{2})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x), x > 0

lo g_{10} (3^{2}) = 2 lo g_{10} (3)

= 2 lo g_{10} (3)

\frac{1}{2} ln (x) - 3 ln (x - 1)

7 ln (x) - \frac{1}{6} ln (y)

ln (\frac{cos ( x )}{cos ( x ) - 1})

What is log_{10}(9x^5)+5log_{10}(1/x) ?
The solution to log_{10}(9x^5)+5log_{10}(1/x) is 2log_{10}(3)