What is ln(1/(e^2))+ln(2e) ?

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

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simplify ln(1/(e^2))+ln(2e)

simplify

ln (\frac{1}{e ^{2}}) + ln (2 e)

ln (2) - 1

Decimal Notation

- 0.30685 \dots

Solution steps

ln (\frac{1}{e ^{2}}) + ln (2 e)

Apply log rule:

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

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

= ln (1) - ln (e^{2}) + ln (2 e)

Apply log rule:

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

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

= ln (1) - ln (e^{2}) + ln (2) + ln (e)

Apply log rule:

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

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

= ln (1) - 2 ln (e) + ln (2) + ln (e)

Group like terms

= - 2 ln (e) + ln (e) + ln (1) + ln (2)

Add similar elements:

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

= - ln (e) + ln (1) + ln (2)

Apply log rule:

lo g_{a} (a) = 1

= - 1 + ln (1) + ln (2)

Apply log rule:

lo g_{a} (1) = 0

= - 1 + 0 + ln (2)

Add/Subtract the numbers:

- 1 + 0 = - 1

= ln (2) - 1

\frac{1}{3} [2 ln (x + 5) - ln (x) - ln (x^{2} - 4)]

lo g_{3} (7) + lo g_{3} (5)

\frac{1}{2} lo g_{6} (9) + lo g_{6} (5)

2 lo g_{b} (y) + 6 lo g_{b} (z)

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