What is ln(1/2 e^{10k}) ?

The solution to ln(1/2 e^{10k}) is 10k-ln(2)

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simplify ln(1/2 e^{10k})

simplify

ln (\frac{1}{2} e^{10 k})

10 k - ln (2)

Solution steps

ln (\frac{1}{2} e^{10 k})

Apply log rule:

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

ln (\frac{1}{2} e^{10 k}) = ln (\frac{1}{2}) + ln (e^{10 k})

= ln (\frac{1}{2}) + ln (e^{10 k})

Apply log rule:

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

ln (e^{10 k}) = 10 k ln (e)

= ln (\frac{1}{2}) + 10 k ln (e)

Apply log rule:

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

ln (\frac{1}{2}) = ln (1) - ln (2)

= ln (1) - ln (2) + 10 ln (e) k

ln (1) = 0

10 k ln (e) = 10 k

= 0 - ln (2) + 10 k

0 - ln (2) + 10 k = - ln (2) + 10 k

= 10 k - ln (2)

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