Expert: Paul Klarreich - 3/2/2007

Question
1.y= log (square root of) 2x+5
2.y= ln (x+3)^4
3.Y= ln (sec x+ tanx)
4.y= ln x-5/x-4

Answer
Questioner:   alijon
Category:  Calculus
Private:  No
 
Subject:  differentation of logaritms functions
Question:  1.y= log (square root of) 2x+5
2.y= ln (x+3)^4
3.Y= ln (sec x+ tanx)
4.y= ln x-5/x-4
..............................
Hi, Ali,

Use these properties of logarithms to simplify:

Ln(AB) = ln A + ln B  [Log of a product = sum of the logs.]
ln(A/B) = ln A - ln B
ln(A^n) = n ln A
sqrt(A) = A^1/2, so  ln (sqrt(A)) = 1/2 ln A

Your examples:

1.y= ln (sqrt(2x+5)) = 1/2 ln(2x + 5)

dy/dx = 1/2 (1/(2x + 5))(2)  << second '2' from the Chain rule.

dy      1
-- = ------
dx   2x + 5

2.y= ln (x+3)^4 = 4 ln(x + 3)

dy     4
-- = ------
dx   x + 3

3.Y = ln (sec x + tanx)

This is a good one to remember.  When you are done, file it away in a good spot for future reference.

Use the Chain rule:  y = ln u,  u = sec x + tan x

dy/dy = 1/u

du/dx = sec x tan x + sec^2(x)

dy   sec x tan x + sec^2(x)
-- = ----------------------
dx       sec x + tan x

Now, a bit of simpifying:

dy   sec x (tan x + sec x)
-- = ----------------------
dx       sec x + tan x

= sec x

WOW! what a simplification!  Some day you will have to integrate  sec x, and then it will be important to remember this example.


4.y= ln x-5/x-4 = ln(x - 5) - ln(x - 4)

dy     1       1
-- = ----- - -----
dx   x - 5   x - 4

Combine over an LCD.  [It's good practice, even if you never see it again, which you will when you study integration by Partial Fractions.]

dy   x - 4 - (x - 5)
-- = ---------------
dx   (x - 5)(x - 4)

dy   x - 4 - x + 5
-- = --------------
dx   (x - 5)(x - 4)

dy        1
-- = -------------
dx   x^2 - 9x + 20