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Contents
Logarithmic Function
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Logarithmic Function 
Derivative
of the Logarithmic Function
Given y = ln x,
In general, for y = ln f(x),
The
derivative of the logarithmic function can be derived using first principles.
However, again, it would involve limits which are hard to evaluate. The
alternative is to use the already-proven derivative of the exponential
function to prove the derivative of ln x.
Let
y = ln x.
Thus,
x = ey
conversion
from logarithmic to exponential form
To
differentiate logarithms other than the natural logarithm, simply change the
base of the logarithm to e, using the change of base formula.
Examples
 Differentiate
with respect to x:
(i)
y = ln (2x + 1)
(ii) y = ln (x2 + 4x + 7)
(iii) y = ln (x - 1)2
(iv)
(x) y = log3 (2x + 3)2
Solutions:
(i)
(ii)
(iii)
This expression can be simplified:
y = ln (x - 1)2
= 2 ln (x - 1)
(iv)
For this question, it is difficult and tedious to differentiate the expression
directly. However, using the properties of logarithms, we can simplify the
differentiation.
= ln (x + 4) - ln (2x + 1)
(x)
Use change of base formula:
y = log3 (2x + 3)2
= 2 log3 (2x + 3)
= 2/ln 3 [ln (2x + 3)]
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