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Differentiation of Logarithmic Functions
The rule for finding the derivative of a logarithmic function is given as:
If log a
y = x then
ln
dy
or y
dx a x
This rule can be proven by rewriting the logarithmic function in exponential form and then using
the exponential derivative rule covered in the last section.
log a
y = x Begin with logarithmic function
Convert into exponential form
y a = x
( ln ) 1 Differentiate both sides of the equation
y a ⋅ a ⋅ y ′=
Substitute x for the exponential function a
y
( ln^ a^ )⋅^ x^ ⋅^ y ′=^1
ln
y
a x
′ = Solve for y´ by dividing each side by “(ln a)x”
As with the exponential rules, the derivative of a logarithmic function can be simplified if the
base of the logarithm is “e”.
log ln e
y = x = x Logarithmic function with base “e”
ln
y
e x
′ = Apply the logarithm derivative rule above
y
x
′ = Use properties of logarithms to simplify
y
x
Example 1: Find the derivative of ( )
6
f x =log x
Solution:
6
log
ln 6
f x x
f x
x
Example 2: Find the derivative of ( )
2 f x =log 4 x
Solution:
Right now the only derivative rule we have for logarithms is for “log x”. So in
order to take the derivative of this function we must first use the properties of
logarithms to rewrite the function.
First use the product property of log ( xy )= log x + log y
2
2
log 4
log 4 log
f x x
x
Next use the exponent property of
( ) (^ )
log log
r x = r ⋅ x
2 log 4 log
log 4 2 log
f x x
x
Now we can find the derivative.
log 4 2 log
ln
ln
f x x
f x
x
x
Even though we were able to find the derivative of this function, it would be easier to find the
derivative if we had a rule that dealt with the situation where a function is equal to the log of
another function. In order to do this, we would have to combine the chain rule with the
logarithm rule.
Example 3: Find the derivative of ( )
2 f x =log 4 x
Solution:
Note: Remember if a base is not shown it is understood to be 10.
2
2
2
2
log 4
ln10 4
ln10 4
ln
ln
x
f x x
D x
f x
x
x
x
x
x
As you can see we get the same derivative as before but in an shorter and easier
process.
Example 4: Find the derivative of
3 4 4 y = ln t + 5 t
Solution:
In this problem we can simplify the derivative process by first applying the
properties of logarithms to move the exponent out in front of the function as a
coefficient.
3 4 4
4
ln 5
ln 5
y t t
t t
Now we can find the derivative
4
ln 5
y = t + t
4
ln 5
x
y D t t
′ = ⎡^ ⎤
Example 4 (Continued):
4
4
x
D t t
y
t t
3
4
t
y
t t
Now lets look at a more complex function that will require the use of several derivative rules.
Remember when finding the derivative of a complex function take it step by step. Don’t try to
do it all at once.
Example 5: Find the derivative of
2 2 1
ln 5 3
x x e
y
x
Solution:
Since the function is in the form of a fraction we must begin by applying the
quotient rule. When you go to find the derivative of the numerator you are will
have to use both the product and exponential rules. The derivative of the
denominator will require the use of the logarithm rule.
First, apply the quotient rule.
2 2
2 2 2 2
2
ln 5 3
ln 5 3 1 1 ln 5 3
ln 5 3
x
x x
x x
x e
y
x
x D x e x e D x
y
x
Example 5 (Continued):
( ) (^ )^ ( )
( )(^ )^ (^ )^ ( )
2 2 2 2 2
2
2 2 2 2
2
2 2 2 2
2
2 2 2
2
ln 5 3 1 2 2 1
ln 5 3
2 1 ln 5 3 1
ln 5 3
2 1 5 3 ln 5 3 1 5
5 3 ln 5 3
2 1 5 3 ln 5 3 5 1
5 3 ln 5 3
x x x
x x
x x
x
x x e xe x e
x
y
x
e x x x x e
x
x
e x x x x x e
x x
e x x x x x
x x
+ ⋅ ⎡^ + + ⎤− + ⋅
