What is ln(1/(e y/x)) ?

The solution to ln(1/(e y/x)) is -ln(y/x)-1

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simplify ln(1/(e y/x))

simplify

ln (\frac{1}{e \frac{y}{x}})

- ln (\frac{y}{x}) - 1

Solution steps

ln (\frac{1}{e \frac{y}{x}})

Apply log rule:

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

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

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

Apply log rule:

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

ln (e \frac{y}{x}) = ln (e) + ln (\frac{y}{x})

= ln (1) - (ln (\frac{y}{x}) + ln (e))

Apply log rule:

lo g_{a} (1) = 0

= 0 - (ln (\frac{y}{x}) + ln (e))

Apply log rule:

lo g_{a} (a) = 1

= 0 - (ln (\frac{y}{x}) + 1)

0 - (1 + ln (\frac{y}{x})) = - (1 + ln (\frac{y}{x}))

= - (ln (\frac{y}{x}) + 1)

Distribute parentheses

= - (ln (\frac{y}{x})) - (1)

Apply minus-plus rules

+ (- a) = - a

= - ln (\frac{y}{x}) - 1

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