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

The solution to ln((e^2)/(e^3))+2ln(3) is -1+2ln(3)

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

simplify

ln (\frac{e ^{2}}{e ^{3}}) + 2 ln (3)

- 1 + 2 ln (3)

Decimal Notation

1.19722 \dots

Solution steps

ln (\frac{e ^{2}}{e ^{3}}) + 2 ln (3)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Add similar elements:

2 ln (e) - 3 ln (e) = - ln (e)

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

Apply log rule:

lo g_{a} (a) = 1

= - 1 + 2 ln (3)

5 ln (2) - 2 ln (\frac{7}{2})

ln (2022) - ln (2020)

2 ln (a) + 5 ln (b) - 7 ln (c)

- \frac{1}{2} ln (x) + ln (x^{- \frac{1}{2}})

ln (\frac{115}{100})

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