What is ln((1/2 x)/x) ?

The solution to ln((1/2 x)/x) is -ln(2)

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simplify ln((1/2 x)/x)

simplify

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

- ln (2)

Decimal Notation

- 0.69314 \dots

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Add similar elements:

ln (x) - ln (x) = 0

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

Apply log rule:

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

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

= ln (1) - ln (2)

Apply log rule:

lo g_{a} (1) = 0

= 0 - ln (2)

0 - ln (2) = - ln (2)

= - ln (2)

2 ln (2 t)

ln (\frac{9993000}{7000})

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

ln (\frac{315}{90})

lo g_{4} (e) - lo g_{4} (5 e) - lo g_{4} (4 e)