5128_Ch03_pp098-184

Section 3.9 Derivatives of Exponential and Logarithmic Functions 175
Setting these two formulas for m equal to each other, we have
ln a
1 a a
ln a 1
e ln a e 1
a e
m 1 . Now try Exercise 31.
e
Derivative of log a x
To find the derivative of log a x for an arbitrary base a 0, a 1, we use the changeof-base
formula for logarithms to express log a x in terms of natural logarithms, as follows:
ln
x
log a x .
l n a
The rest is easy:
d d
log d x a x d x ( ln
ln
x
) a
1 d
• ln x
ln a d x
Since ln a is a constant
1
• 1 ln a x
1
.
x l n a
So, if u is a differentiable function of x and u 0, the formula is as follows.
For a 0 and a 1,
d
1
log d x a u d u
. u l n a dx
EXAMPLE 4 Going the Long Way with the Chain Rule
Find dydx if y log a a sin x .
SOLUTION
Carefully working from the outside in, we apply the Chain Rule to get:
d
log d x a a sin x 1 d
a
sin x • a
ln a d x
sin x
1
a
sin x • a
ln a
sin x d
ln a • sin x d x
log a u, u a sin x
a u , u sinx
a sin
x
ln
a
as
in
x • cos x
ln
a
cos x. Now try Exercise 23.