What is ln(x^5)-ln(x)+ln(1/x) ?

The solution to ln(x^5)-ln(x)+ln(1/x) is ln(x^3)

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simplify ln(x^5)-ln(x)+ln(1/x)

simplify

ln (x^{5}) - ln (x) + ln (\frac{1}{x})

ln (x^{3})

Solution steps

ln (x^{5}) - ln (x) + ln (\frac{1}{x})

Apply log rule:

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

ln (x^{5}) - ln (x) = ln (\frac{x ^{5}}{x})

= ln (\frac{x ^{5}}{x}) + ln (\frac{1}{x})

Cancel the common factor:

x

= ln (x^{4}) + ln (\frac{1}{x})

Apply log rule:

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

ln (x^{4}) + ln (\frac{1}{x}) = ln (\frac{1}{x} x^{4})

= ln (\frac{1}{x} x^{4})

Simplify

x^{4} \frac{1}{x} : x^{3}

= ln (x^{3})

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

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

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

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

f (x) = lo g_{3} (27 x)

What is ln(x^5)-ln(x)+ln(1/x) ?
The solution to ln(x^5)-ln(x)+ln(1/x) is ln(x^3)