What is ln((10^{-4}x^{-2})/(x^5)) ?

The solution to ln((10^{-4}x^{-2})/(x^5)) is -4ln(10)-7ln(x)

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simplify ln((10^{-4}x^{-2})/(x^5))

simplify

ln (\frac{1 0 ^{- 4} x ^{- 2}}{x ^{5}})

- 4 ln (10) - 7 ln (x)

Solution steps

ln (\frac{1 0 ^{- 4} x ^{- 2}}{x ^{5}})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{1 0 ^{- 4} x ^{- 2}}{x ^{5}}) = ln (1 0^{- 4} x^{- 2}) - ln (x^{5})

= ln (1 0^{- 4} x^{- 2}) - ln (x^{5})

Apply log rule:

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

ln (x^{5}) = 5 ln (x)

= ln (1 0^{- 4} x^{- 2}) - 5 ln (x)

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (1 0^{- 4} x^{- 2}) = ln (1 0^{- 4}) + ln (x^{- 2})

= ln (1 0^{- 4}) + ln (x^{- 2}) - 5 ln (x)

Apply log rule:

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

ln (1 0^{- 4}) = - 4 ln (10)

= - 4 ln (10) + ln (x^{- 2}) - 5 ln (x)

Apply log rule:

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

ln (x^{- 2}) = - 2 ln (x)

= - 4 ln (10) - 2 ln (x) - 5 ln (x)

Add similar elements:

- 2 ln (x) - 5 ln (x) = - 7 ln (x)

= - 4 ln (10) - 7 ln (x)

- \frac{2}{5} lo g_{2} (\frac{2}{5}) - \frac{3}{5} lo g_{2} (\frac{3}{5})

lo g_{10} (\frac{x}{10})

- \frac{3}{5} lo g_{2} (\frac{3}{5}) - \frac{2}{5} lo g_{2} (\frac{2}{5})

lo g_{3} (27 x)

lo g_{5} (\frac{4}{x ^{2} 1 - x})

What is ln((10^{-4}x^{-2})/(x^5)) ?
The solution to ln((10^{-4}x^{-2})/(x^5)) is -4ln(10)-7ln(x)