4 Condense the expression to the logarithm of a single quant

4. Condense the expression to the logarithm of a single quantity.

Solution

given

1/2 [ ln(x+1) + 5ln(x-1) + 2(6 ln x - ln sqrt(x) ]

first a lnb = ln b^a formula

so 6 lnx = ln x^6 and 5 ln(x-1) = ln(x-1)^5

1/2 [ ln(x+1) + ln(x-1)^5 + 2( ln x^6 - ln sqrt(x) ]

now ln a - ln b = ln(a/b)

so  ln x^6 - ln sqrt(x) = ln x^6 / sqrt(x)

1/2 [ ln(x+1) + ln(x-1)^5 + 2( ln x^6 / sqrt(x) ]

now 2( ln x^6 / sqrt(x) ] = ln x^12 / x

1/2 [ ln(x+1) + ln(x-1)^5 + ln x^12 / x ]

1/2 [ ln(x+1) + ln(x-1)^5 + ln x^11 ]

now ln a + ln b+ ln c = log abc

so 1/2 ln (x+1) (x-1)^5 x^11) = [ (x+1) (x-1)^5 x^11) ] ^1/2