simplify ln(1/(2x))

Solution

simplify

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

- ln (x) - ln (2)

Solution steps

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

Apply log rule:

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

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

= ln (1) - ln (2 x)

Apply log rule:

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

ln (2 x) = ln (2) + ln (x)

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

Apply log rule:

lo g_{a} (1) = 0

= 0 - (ln (x) + ln (2))

0 - (ln (2) + ln (x)) = - (ln (2) + ln (x))

= - (ln (x) + ln (2))

Distribute parentheses

= - (ln (x)) - (ln (2))

Apply minus-plus rules

+ (- a) = - a

= - ln (x) - ln (2)

lo g_{4} (9 x)

lo g_{10} (\frac{x y ^{5}}{z ^{5}})

lo g_{10} (x^{2} \frac{x + 3}{x - 1})

\frac{1}{2} lo g_{3} (9 x) + \frac{3}{2} lo g_{3} (4 x^{3})

lo g_{12} (\frac{12}{y})