What is ln(e^2ln(e^3)) ?

The solution to ln(e^2ln(e^3)) is ln(3)+2

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simplify ln(e^2ln(e^3))

simplify

ln (e^{2} ln (e^{3}))

ln (3) + 2

Decimal Notation

3.09861 \dots

Solution steps

ln (e^{2} ln (e^{3}))

Apply log rule:

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

ln (e^{2} ln (e^{3})) = ln (e^{2}) + ln (ln (e^{3}))

= ln (e^{2}) + ln (ln (e^{3}))

Apply log rule:

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

ln (e^{2}) = 2 ln (e)

= 2 ln (e) + ln (ln (e^{3}))

Apply log rule:

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

ln (e^{3}) = 3 ln (e)

= 2 ln (e) + ln (3 ln (e))

Apply log rule:

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

ln (3 ln (e)) = ln (3) + ln (ln (e))

= 2 ln (e) + ln (3) + ln (ln (e))

2 ln (e) = 2

ln (ln (e)) = 0

= 2 + ln (3) + 0

Add the numbers:

2 + 0 = 2

= ln (3) + 2

\frac{1}{2} (ln (x - 1) - ln (3))

\frac{1}{3} ln (\frac{5}{2})

3 lo g_{4} (x + 1) - lo g_{4} (x)

ln (x^{- 3}) + ln (x^{2})

ln (\frac{a}{b}) - 2 ln (a^{3}) + 4 ln (b)