What is ln(e^{in}(e^{in(ex)})) ?

The solution to ln(e^{in}(e^{in(ex)})) is i(n+enx)

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simplify ln(e^{in}(e^{in(ex)}))

simplify

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

i (n + e n x)

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

= in ln (e) + in e x ln (e)

in ln (e) = in

in e x ln (e) = i e n x

= in + i e n x

Rewrite

in + in e x

in standard complex form:

(n + n e x) i

= (n + n e x) i

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What is ln(e^{in}(e^{in(ex)})) ?
The solution to ln(e^{in}(e^{in(ex)})) is i(n+enx)