What is ln((x^2sqrt(x^2-1))/(e^x)) ?

The solution to ln((x^2sqrt(x^2-1))/(e^x)) is 2ln(x)+ln(sqrt(x^2-1))-x

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simplify ln((x^2sqrt(x^2-1))/(e^x))

simplify

ln (\frac{x ^{2} x ^{2} - 1}{e ^{x}})

2 ln (x) + ln (x^{2} - 1) - x

Solution steps

ln (\frac{x ^{2} x ^{2} - 1}{e ^{x}})

Apply log rule:

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

ln (\frac{x ^{2} x ^{2} - 1}{e ^{x}}) = ln (x^{2} x^{2} - 1) - ln (e^{x})

= ln (x^{2} x^{2} - 1) - ln (e^{x})

Apply log rule:

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

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

= ln (x^{2} x^{2} - 1) - x ln (e)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

x ln (e) = x

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

\frac{1}{3} lo g_{b} (x) + 4 lo g_{b} (y) - 2 lo g_{b} (x)

2 lo g_{10} (2) + 3 lo g_{10} (3) - 2 lo g_{10} (6)

lo g_{10} (\frac{a ^{5} a ^{2} b}{( ab ) ^{2}})

lo g_{10} (y^{5} x)

lo g_{a} (\frac{y ^{5} w ^{2}}{x ^{4} z ^{3}})

What is ln((x^2sqrt(x^2-1))/(e^x)) ?
The solution to ln((x^2sqrt(x^2-1))/(e^x)) is 2ln(x)+ln(sqrt(x^2-1))-x