What is ln(4e^{(2pix)/3}) ?

The solution to ln(4e^{(2pix)/3}) is 2ln(2)+(2pix)/3

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simplify ln(4e^{(2pix)/3})

simplify

ln (4 e^{\frac{2 π x}{3}})

2 ln (2) + \frac{2 π x}{3}

Solution steps

ln (4 e^{\frac{2 π x}{3}})

Apply log rule:

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

ln (4 e^{\frac{2 π x}{3}}) = ln (4) + ln (e^{\frac{2 π x}{3}})

= ln (4) + ln (e^{\frac{2 π x}{3}})

Apply log rule:

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

ln (e^{\frac{2 π x}{3}}) = \frac{2 π x}{3} ln (e)

= ln (4) + \frac{2 π x}{3} ln (e)

\frac{2 π x}{3} ln (e) = \frac{2 π x}{3}

= ln (4) + \frac{2 π x}{3}

ln (4) = 2 ln (2)

= 2 ln (2) + \frac{2 π x}{3}

ln (\frac{1073.15}{273.15})

ln (121) - ln (100)

ln (\frac{19.44}{12.62})

- lo g_{2} (\frac{1}{20})

ln (x) + ln (1000 - x)

What is ln(4e^{(2pix)/3}) ?
The solution to ln(4e^{(2pix)/3}) is 2ln(2)+(2pix)/3