Expert: Paul Klarreich - 5/5/2007

Question
Hi, I need help writing (1/2)In(x^2+1)-4In(1/2)-(1/2)(In(x-4)+In x) as a single logarithm. Thanks!

Answer
Questioner:   Sarah
Category:  Advanced Math
Question:  Hi, I need help writing (1/2)In(x^2+1)-4In(1/2)-(1/2)(In(x-4)+In x) as a single

logarithm. Thanks!
............................
Hi, Sarah,

Use these rules:

I. ln a + ln b = ln (ab)
II. ln a - ln b = ln (a/b)
IIa.  - ln a = ln(1/a)
III. n ln x = ln (x^n)
IIIa.  (1/n) ln x = ln (nth root of x) or  ln(x^(1/n))

--- IIIa is just a variation of rule III

(1/2)Ln(x^2+1) - 4 Ln(1/2) - (1/2)(Ln(x-4) + Ln x)  

Take a couple of pieces of that:

(1/2)Ln(x^2+1) =
ln sqrt(x^2 + 1)

.................

- 4 Ln(1/2) =
4 ln 2 = ln 2^4 = ln 16

...............

- (1/2)(Ln(x-4) + Ln x) =
- (1/2)Ln((x-4)x =
- ln sqrt(x(x-4))

..................................

Put them together:

ln sqrt(x^2 + 1)  PLUS  ln 16 - ln sqrt(x(x-4))

ln [ 16 sqrt(x^2 + 1)/sqrt(x(x-4))]

ln [ 16 sqrt((x^2 + 1)/(x(x-4)) ]

I think that does it.