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

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

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

simplify

ln (- \frac{5}{4} e^{4 x})

ln (5) - 2 ln (2) + 4 x

Solution steps

ln (- \frac{5}{4} e^{4 x})

Apply log rule:

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

ln (- \frac{5}{4} e^{4 x}) = ln (\frac{5}{4}) + ln (e^{4 x})

= ln (\frac{5}{4}) + ln (e^{4 x})

Apply log rule:

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

ln (e^{4 x}) = 4 x ln (e)

= ln (\frac{5}{4}) + 4 x ln (e)

Apply log rule:

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

ln (\frac{5}{4}) = ln (5) - ln (4)

= ln (5) - ln (4) + 4 ln (e) x

4 x ln (e) = 4 x

= ln (5) - ln (4) + 4 x

ln (4) = 2 ln (2)

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

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