What is ln((e^3sqrt(2))/(e^{1/3)})-e^{2ln(2)} ?

The solution to ln((e^3sqrt(2))/(e^{1/3)})-e^{2ln(2)} is -4/3+1/2 ln(2)

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simplify ln((e^3sqrt(2))/(e^{1/3)})-e^{2ln(2)}

simplify

ln (\frac{e ^{3} 2}{e ^{\frac{1}{3}}}) - e^{2 l n (2)}

- \frac{4}{3} + \frac{1}{2} ln (2)

Decimal Notation

- 0.98675 \dots

Solution steps

ln (\frac{e ^{3} 2}{e ^{\frac{1}{3}}}) - e^{2 l n (2)}

Apply log rule:

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

ln (\frac{e ^{3} 2}{e ^{\frac{1}{3}}}) = ln (e^{3} 2) - ln (e^{\frac{1}{3}})

= ln (2 e^{3}) - ln (e^{\frac{1}{3}}) - e^{2 l n (2)}

Apply log rule:

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

ln (e^{\frac{1}{3}}) = \frac{1}{3} ln (e)

= ln (e^{3} 2) - \frac{1}{3} ln (e) - e^{2 l n (2)}

Apply log rule:

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

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

= ln (e^{3}) + ln (2) - \frac{1}{3} ln (e) - e^{2 l n (2)}

Apply log rule:

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

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

= 3 ln (e) + ln (2) - \frac{1}{3} ln (e) - e^{2 l n (2)}

Group like terms

= - \frac{1}{3} ln (e) + 3 ln (e) - e^{2 l n (2)} + ln (2)

Add similar elements:

- \frac{1}{3} ln (e) + 3 ln (e) = \frac{8}{3} ln (e)

= \frac{8}{3} ln (e) - e^{2 l n (2)} + ln (2)

\frac{8}{3} ln (e) = \frac{8}{3}

e^{2 l n (2)} = 2^{2}

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

= \frac{8}{3} - 2^{2} + \frac{1}{2} ln (2)

2^{2} = 4

= \frac{8}{3} - 4 + \frac{1}{2} ln (2)

Combine the fractions

- 4 + \frac{8}{3} : - \frac{4}{3}

= - \frac{4}{3} + \frac{1}{2} ln (2)

- ln (a) - ln (b)

ln (11 x)

\frac{1}{2} lo g_{a} (x) + 3 lo g_{a} (y) - 4 lo g_{a} (x)

lo g_{a} (3 x y) - 5 lo g_{10} (a) (y + 2)

lo g_{3} (4 x y)

What is ln((e^3sqrt(2))/(e^{1/3)})-e^{2ln(2)} ?
The solution to ln((e^3sqrt(2))/(e^{1/3)})-e^{2ln(2)} is -4/3+1/2 ln(2)