Subject: Calculus

Natural Logarithm

What exactly is a logarithm?

In simple words, logarithm (also written as \log) is a function that scales down the value of a number or another function. The scale is respective of another number known as base. What we mean when we take the logarithm of x with respect to the base b (\log_{b}{x}) is the number of times the base b is multiplied with itself to attain the value of x. So if we have for example the logarithm of 100 to the base 10 (\log_{10}{100}), then it would be 2 since there are exactly two factors of 10 to produce 100, thus the logarithm of 100 to the base 10 is 2 (\log_{10}{100}=2). Note also that 100=10\cdot 10 =10^2. So effectively, logarithms can also be put into simple words as "exponent or power to which a number or function can be raised to yield a specific number".

If so, then what is a natural logarithm?

Technically, natural logarithms are simply logarithmic functions just like what we have discussed above. However, natural logarithms are special kinds of logarithms because its base is a natural number (also known as e) hence the word "natural" logarithm. e is a rational positive number whose value is approximately equal to 2.71828183. So now have an identity between logarithm and natural logarithm such that \log_{e}{x}=\ln{x}.



Plot of the function y=\ln{x} in a Euclidean space.


Natural logarithms, like any other ordinary logarithms are very helpful in problem solving that involves equations in which the unknown appears as the exponent of some other quantity. Some very famous calculations that is dealt with logarithms are problems involving half-life, decay constant, or unknown time in exponential decay problems. Not just in science and mathematics but natural logarithms are also used in finance in dealing problems involving compound interest.
The best thing about natural logarithms that is readily not applicable to any ordinary logarithms is that it can be defined easily in simple derivatives, integrals and Taylor series. Also, the fact that its property includes the natural logarithm of the base itself as 1 makes it more interesting compared to any other logarithms. The frequent appearance of e in many mathematical and scientific equations, functions and expressions make the natural logarithm more viable to use than ordinary logarithms.

Great! Now, tell me the basic properties and identities of natural logarithms

Natural logarithms have the following properties as well as identities that can be very useful in many calculations...

Ordinary logarithm to natural logarithm

\log_{e}{x}=\ln{x}

The natural logarithm of the base itself is 1.

\ln{e}=1

The natural logarithm of 1 is zero.

\ln{1}=0

Exponential identities.

e^{\ln{x}}=x \ln{e^{x}}=x

Product

\ln{ab}=\ln{a}+\ln{b}

Quotient

\ln{\frac{a}{b}}=\ln{a}-\ln{b}

Derivative

\frac{d}{dx}\ln{x}=\frac{1}{x}

Indefinite Integral

\int \ln{x}dx = x\ln{x} - x + C


Sample problems (decay problems)

Problems involving natural logarithms includes decay rates of substances. Theoretically, the decay mechanism of any substances is governed by the following formula,

S=S_{o}e^{-kt}

where S_{o} is the initial amount of substance, t is the time, S is the actual amount at time t and k is a constant which is varies upon the substance.

Example #1

Now, given the constant k=\frac{ln{3}}{30}, find the time it takes to convert a substance to half of its value.

t=16.75 seconds

Example #2

Find the time derivative of S.

S=S_oe^{-kt}

Differentiating S with respect to t, we get

D_{t}S=S_oe^{-kt}(-k)

Thus,

\frac{dS}{dt} = -kS_oe^{-kt}

Example #3

Explain why in the following subtraction of natural logarithm,

\ln{a}-\ln{b}

the value of b can never be zero.

At first look, we may argue that b can take any number including zero. Now, let me show why it can never be zero.

Using quotient rule. we convert

\ln{a}-\ln{b}

into

\frac{\ln{a}}{\ln{b}}

Now, if b is zero, we will get an undefined value since we cannot divide be zero. As a matter of fact, all natural logarithms of negative numbers and zero itself is undefined or infinite.

Example #4

Find the derivative of the following logarithmic function:

f(x) = \frac{1}{2}\ln{2x}

f(x) = \frac{1}{2}\ln{2x}

Differentiating, we have

f'(x) = \frac{1}{2}\frac{1}{2x}\cdot 2

f'(x) = \frac{1}{2x}

Therefore:

f'(x) = \frac{1}{2x}

Example #5

Solve for the integral of the logarithmic function below.

\int \ln{e}dx

The integrand is a logarithmic function but is a constant function, so we easily integrate.

\int \ln{e}dx=\ln{e} \int dx

=x\ln{e}

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