Subject: Calculus

Logarithmic Derivative

Logarithms (log) are exponent or power to which a number or function, can be raised to yield a specific number. We call the number being raised as the "base". Logarithms can also be referred to as "mathematical shortcut" for exponents. When you have a number say, 1000, then the logarithm of that number with respect to the base, say 10, is 3, which is written as

\log_{10}{1000} = 3

Why do I say shortcut? Because as you see , the number 1000 can be written as the multiplication of three (3) 10's (1000=10^3), which in this case is the base. So imagine if you're working with large numbers (say, trillionth degree). Then it would be laborious to oneself writing down such big amounts of numbers. That's where the logarithm takes place. So instead of writing down the whole thing with the actual large numbers, you can use the logarithmic scale and take down the logarithms of all these large numbers you're working with. Then, life would be easier.
On the other hand, when a function f is to take its logarithm, then it is written as \log{f}.

Basic Properties of Logarithm

Properties of logarithm are also called as "logarithmic identities". This covers the basic mathematical operations for logarithms.

Product

The logarithm of a product is equal to the sum of the two logarithms:
\log_{x}{ab}=\log_{x}{b}+\log_{x}{a}

Quotient

The logarithm of a quotient is equal to the difference of the two logarithms:
\log_{x}\frac{a}{b}=\log_{x}{b} - \log_{x}{a}

Power

The logarithm of the n-th power of a number is n times the logarithm of that number:
\log_{x}{a^n}=n\log_{x}{a}

Differentiation of Logarithmic Function

The derivative of the logarithm of a function (\log{f} is defined as the logarithmic derivative of a function. The formula of the logarithmic derivative is given by;

D_{x}[\log{f}] = \frac{f'}{f}

So basically, logarithmic derivative is just as easy as ordinary derivatives. The only difference is that in logarithmic derivative, we have to divide the derivative of a function by the function itself. For example, we have a function f(x)=x^2, then its logarithmic derivative is given by;

f(x)=x^2
f'(x)=2x

Thus,

\frac{f'(x)}{f}=\frac{2x}{x^2}=\frac{2}{x}

Properties of Logarithmic Derivative

We have discussed above the properties of logarithm which in turn, has its analog properties in logarithmic derivative. They are as follows;

Product

The logarithmic derivative of a product of two functions is equal to the sum of each individual logarithmic derivatives.:
\frac{\left (ab\right )'}{ab}= \frac{a'}{a}+\frac{b'}{b}

Quotient

The logarithmic derivative of a ratio of two functions is the difference of their individual logarithmic derivatives.:
\frac{\left (\frac{a}{b}\right )'}{\frac{a}{b}}= \frac{a'}{a}-\frac{b'}{b}

Power

The logarithmic derivative of the n-th power of a function is n-times the logarithmic derivative of that function:
\frac{\left (f^{p}\right )'}{f}=p \frac{f'}{f}

Standard Notations

When the base of a logarithm is 10, then the subscript that indicates the base is conventionally omitted. Thus, expressions such as \log{x}, \log{f}, \log{a} means that the logarithm has the base 10.
Furthermore, another form of logarithm is the natural logarithm written as \ln{x}. Instead of the word log, natural logarithm uses ln. The only difference between logarithm and natural logarithm are its base. While logarithms cover all values for the base, natural logarithms indicates a base of e only where e is a dimensionless natural number whose value is 2.71828183. Additionally, natural logarithms are assigned only to a range of non-zero and below numbers. In short, there is no natural logarithm of negative numbers and zero.

Example #1

What is the logarithmic derivative of the function f=x+1?

D_x\log{f}=\frac{1}{x+1}

Example #2

Find the logarithmic derivative when the function is f(u)=\cos{u}.

D_u\log{f(u)}=\frac{-\sin{u}}{\cos{u}} = -\tan{u}

Example #3

Show that the logarithmic derivative of the function f(x)=e^x is one.

First, we have to get the derivative of f,

f'(x)=e^x.

Then, we divide the derivative of f by f itself, that is

\frac{f'(x)}{f(x)} = \frac{e^x}{e^x}=1

Thus,

D_x\log{e^x}=1

Example #4

Using the product property of logarithmic derivative, find the logarithmic derivative of f(s)=e^x\sin{x}.

Product property of logarithmic derivative states that

\frac{\left (ab\right )'}{ab}= \frac{a'}{a}+\frac{b'}{b}

In this case, we can assign e^x=a and \sin{x}=b. Then we find the individual derivatives of a and b,

a' = e^x

b'=\cos{x}

Then,

\frac{a'}{a}+\frac{b'}{b}= \frac{e^x}{e^x}+\frac{\cos{x}}{\sin{x}}

Therefore,

D_s\log{e^x\sin{x}} = 1+\cot{x}

Example #5

Find the logarithmic derivative of a logarithmic function. That is D[\log{\log{f}}]

D[\log{\log{f}}] = \frac{f'}{(\log{f})^2}

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