What is ln(1/8 e^{8x}) ?

The solution to ln(1/8 e^{8x}) is 8x-3ln(2)

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simplify ln(1/8 e^{8x})

simplify

ln (\frac{1}{8} e^{8 x})

8 x - 3 ln (2)

Solution steps

ln (\frac{1}{8} e^{8 x})

Apply log rule:

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

ln (\frac{1}{8} e^{8 x}) = ln (\frac{1}{8}) + ln (e^{8 x})

= ln (\frac{1}{8}) + ln (e^{8 x})

Apply log rule:

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

ln (e^{8 x}) = 8 x ln (e)

= ln (\frac{1}{8}) + 8 x ln (e)

Apply log rule:

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

ln (\frac{1}{8}) = ln (1) - ln (8)

= ln (1) - ln (8) + 8 ln (e) x

ln (1) = 0

8 x ln (e) = 8 x

= 0 - ln (8) + 8 x

0 - ln (8) + 8 x = - ln (8) + 8 x

= 8 x - ln (8)

ln (8) = 3 ln (2)

= 8 x - 3 ln (2)

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