3.1 Definition of logarithm 3.2 Laws of logarithm 3.3 simple examples based on laws

1. Logarithm
By GP - Mumbai

2. Contents:
3.1 Definition of logarithm
3.2 Laws of logarithm
3.3 simple examples based on laws

3. 1. Introduction
• Logarithmic functions are the inverses of exponential functions, and any
exponential function can be expressed in logarithmic form.
• Similarly, all logarithmic functions can be rewritten in exponential form.
• Logarithms are really useful in permitting us to work with very large
numbers while manipulating numbers of a much more manageable size.
• Every exponential function f(x) = a x, with a > 0 and a ≠ 1. is a one-to-one
function, therefore has an inverse function
(f-1).
• The inverse function is called the Logarithmic function with base a and is
denoted by loga

4. Use (Applications) of logarithm in real life:
1. The magnitude of an earthquake is a Logarithmic scale. The famous "Richter Scale" uses this
formula:
Where A is the amplitude (in mm) measured by the Seismograph and B is a distance
correction factor.
2. Loudness is measured in Decibels (dB for short):
Loudness in Where p is the sound pressure.
3. Acidity (or Alkalinity) is measured in pH:
Where H+ is the molar concentration of dissolved hydrogen ions.

5. Definition of logarithm
• Let y be a positive number with. The logarithmic function with
base a , Denoted by loga is defined by:
Clearly, is the exponent to which the base a must be raised to
give.
• For example:
Exponential form
Logarithm form

6. Inverse of the logarithm
ax = y is the inverse of logay = x
Exponential form:
Logarithmic form:

7. Laws of indices:

8. Laws of Logarithm Part -1
1. The first law of logarithms (Product law of logarithm)
2. The second law of logarithms (Power law of logarithm)
3. The third law of logarithms (Quotient law of logarithm)

9. Laws of Logarithm Part -2
4. Rule of Change of base
5. The logarithm of 1
Recall that any number raised to the power zero is 1,. The logarithmic form of this is
6. Logarithm of a to the base a
As then its equivalent logarithm form is

10. 1.The first law of logarithms (Product law of logarithm)
Statement:
Proof:
Suppose
Then the equivalent logarithmic forms are
Using the first rule of indices
Now the logarithmic form of the statement
So, if we want to multiply two numbers together and find the logarithm of the result, we can do this by adding together the logarithms of the two numbers. This
is the first law.

11. 2. The second law of logarithms (Power law of logarithm)
Statement:
Proof:
Suppose
Or equivalently
Suppose we raise both sides of to the power
Using the rules of indices we can write this as
Thinking of the quantity as a single term, the logarithmic form is
This is the second law. It states that when finding the logarithm of a power of a number, this can be evaluated by multiplying the logarithm
of the number by that power.

12. 3. The third law of logarithms (Quotient law of logarithm)
Statement:
Proof: As before, suppose
Then the equivalent logarithmic forms are
Consider.
Using the second rule of indices
Now the logarithmic form of the statement

13. 4. Rule of Change of base
Statement:
Proof: Suppose
And
From (1) the equivalent logarithm form is
From (2), we can write the equivalent exponential form as
Bases are same on both sides of equality, that means powers are same, on equating we get,
Hence proved

14. THANK YOU !