What is ln((5ln(L)(2ln(M)))/(1/2 ln(K))) ?

The solution to ln((5ln(L)(2ln(M)))/(1/2 ln(K))) is ln(5)+2ln(2)+ln(ln(L))+ln(ln(M))-ln(ln(K))

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simplify ln((5ln(L)(2ln(M)))/(1/2 ln(K)))

simplify

ln (\frac{5 ln ( L ) ( 2 ln ( M ) )}{\frac{1}{2} ln ( K )})

ln (5) + 2 ln (2) + ln (ln (L)) + ln (ln (M)) - ln (ln (K))

Solution steps

ln (\frac{5 ln ( L ) ( 2 ln ( M ) )}{\frac{1}{2} ln ( K )})

Apply log rule:

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

ln (\frac{5 ln ( L ) ( 2 ln ( M ) )}{\frac{1}{2} ln ( K )}) = ln (5 ln (L) (2 ln (M))) - ln (\frac{1}{2} ln (K))

= ln (5 \cdot 2 ln (L) ln (M)) - ln (\frac{1}{2} ln (K))

Apply log rule:

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

ln (5 ln (L) \cdot 2 ln (M)) = ln (5) + ln (ln (L)) + ln (2) + ln (ln (M)), ln (\frac{1}{2} ln (K)) = ln (\frac{1}{2}) + ln (ln (K))

= ln (5) + ln (ln (L)) + ln (2) + ln (ln (M)) - (ln (ln (K)) + ln (\frac{1}{2}))

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= ln (5) + ln (ln (L)) + ln (2) + ln (ln (M)) - (ln (ln (K)) - ln (2))

- (- ln (2) + ln (ln (K))) : ln (2) - ln (ln (K))

= ln (5) + ln (ln (L)) + ln (2) + ln (ln (M)) + ln (2) - ln (ln (K))

Simplify

ln (5) + ln (ln (L)) + ln (2) + ln (ln (M)) + ln (2) - ln (ln (K)) : ln (5) + 2 ln (2) + ln (ln (L)) + ln (ln (M)) - ln (ln (K))

= ln (5) + 2 ln (2) + ln (ln (L)) + ln (ln (M)) - ln (ln (K))

ln (x) + 3 ln (y) - 5 ln (z)

lo g_{2} (\frac{p ^{3} q}{16 r ^{2}})

\frac{1}{3} ln (\frac{65}{76})

3 ln (x) + 4 ln (yz) - ln (x^{2} y^{2})

3 ln (3) - 4 ln (2) + 1

What is ln((5ln(L)(2ln(M)))/(1/2 ln(K))) ?
The solution to ln((5ln(L)(2ln(M)))/(1/2 ln(K))) is ln(5)+2ln(2)+ln(ln(L))+ln(ln(M))-ln(ln(K))