What is ln(t^2x^αy^β) ?

The solution to ln(t^2x^αy^β) is 2ln(t)+αln(x)+βln(y)

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simplify ln(t^2x^αy^β)

simplify

ln (t^{2} x^{α} y^{β})

2 ln (t) + α ln (x) + β ln (y)

Solution steps

ln (t^{2} x^{α} y^{β})

Apply log rule:

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

ln (t^{2} x^{α} y^{β}) = ln (t^{2}) + ln (x^{α}) + ln (y^{β})

= ln (t^{2}) + ln (x^{α}) + ln (y^{β})

Apply log rule:

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

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

= 2 ln (t) + ln (x^{α}) + ln (y^{β})

Apply log rule:

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

ln (x^{α}) = α ln (x)

= 2 ln (t) + α ln (x) + ln (y^{β})

Apply log rule:

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

ln (y^{β}) = β ln (y)

= 2 ln (t) + α ln (x) + β ln (y)

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