What is ln(1/(2e^{x^2)}) ?

The solution to ln(1/(2e^{x^2)}) is -x^2-ln(2)

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simplify ln(1/(2e^{x^2)})

simplify

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

- x^{2} - ln (2)

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

lo g_{a} (1) = 0

= 0 - (ln (e) x^{2} + ln (2))

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

= 0 - (x^{2} + ln (2))

0 - (ln (2) + x^{2}) = - (ln (2) + x^{2})

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

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