DERIVATIVES OF lOGARITHMIC FUNCTIONS


If a is a positive real number other than 1, then the graph of the exponential function with base a passes the horizontal line test. In this case, the inverse of the exponential function with base a is called the logarithmic function with base a, and is denoted loga(x). In particular, the natural logarithm is the logarithmic function with base e.

ln(e) = loge(x)

The graphs of two other logarithmic functions are displayed below.
Logarithmic Functions with Base 2 and 1/2
Graphs

Here is the general result regarding differentiation of logarithmic functions. The base a is any fixed positive real number other than 1.

Derivatives of Logarithmic Functions
d

dx
loga(x) =
1

xln(a)


There are at least two ways to verify this differentiation formula.

Approach #1. Express loga(x) in terms of ln(x): loga(x) = ln(x)/ln(a).

Let u = loga(x). This means that au = x. Now take the natural log of both sides.

ln(au) = ln(x)

We then apply an algebraic property of the logarithm and solve for u.

uln(a) = ln(x)

loga(x) = u = ln(x)/ln(a)


Thus, the logarithm base a is just a constant multiple of the natural logarithm. Knowing the derivative of the natural log, the result follows from the linearity of the derivative.

Dx(loga(x)) = Dx(ln(x)/ln(a)) = [1/ln(a)] Dx(ln(x)) = [1/ln(a)][1/x] = 1/xln(a)

Approach #2. Use the chain rule: alog a(x) = x.

alog a(x) = x

Dx(alog a(x)) = Dx(x)

[aloga(x)(ln(a))] Dx(log a(x)) = 1

Dx(loga(x)) = 1/alog a(x)(ln(a)) = 1/xln(a)



Combining the derivative formula for logarithmic functions, we record the following formula for future use. Here, a is a fixed positive real number other than 1 and u is a differentiable function of x.

Derivatives of Logarithmic Functions
d

dx
loga(u) =
1

uln(a)
du

dx

This concludes the Lesson for Stage 6. You should go to the Practice area and .... Practice!! There are many new computational techniques here. You will find that you can use them quite easily if you give yourself a chance.


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William A. Bogley
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