Mathematicians use the logarithm to express the inverse function of exponentiation. That is, the logarithm of a given number p is the exponent to which another fixed number, the base b, must be raised in order to obtain that number p. In the simplest case, the logarithm counts how many times the same factor appears in repeated multiplication;

In the simplest case, the logarithm is defined as the number of times we multiply one number to get another number.

e.g. 625 = 25 × 25 × 25= 25^{3}, the “logarithm base 25” of 625 is 3, or log_{25}(625) = 3. Sound, earthquakes (Richter scale), star brightness, and chemistry are a few examples where logarithm is used.

A logarithm describes how a number must be multiplied to obtain additional values. It is the most practical method of expressing huge numbers. Several important properties of logarithms enable them to be expressed as additions and subtractions of logarithms.

The exponent by which ” b” must be raised to get “ a “is the logarithm of a positive real number “a “ with respect to base” b”, a positive real number not equal to 1.

in other words,

**b _{y}= a ; log_{b}a=y**

Where “a” and “b” are two real numbers that are both positive.

The number y is a real number.

The argument “a” is located within the log, and the base “b” is found at the bottom.

### Properties of Logarithm

- The sum of the logarithms of two numbers says a and b, is the logarithm of the product of the two numbers. Both numbers must have the same base. In other words, this theorem is known as the “product rule for logarithms”.

For example log_{a }(3x) = log_{a }(x) + log_{a }(3)

We cannot use the property of the product to simplify a logarithm which is log_{3}(8) + log_{4}(x) - The difference of the logarithms of two numbers, such as a and b, is equal to the logarithm of their division. Even in this case both numbers should have the same base. There is another name for this theorem, which is the “quotient rule for logarithms”.

For example log_{5}= log_{5}6 – log_{5}7

Similarly log_{5}8 – log_{5}9 = log_{5} - We know from this property that the power of the log is the exponent times the logarithm of the base.

For example

Solving log_{4}(x^{3}) ,

By using the power of logarithm property converts the single logarithm to multiple logarithms.

log_{4}(x^{3})=3 log_{4}(x)

We can also convert a multiple of a logarithm into a single logarithm. We can take the help of the quotient rule of logarithm to do this

For example

4 log_{4}(8) = log_{4}(8^{4}) = log_{4}(4096)

### Applications of Logarithm

Most applications that involve the concept of exponentials use logarithms for calculations.

- The magnitude of earthquakes can easily be measured using
- They are used for measuring the noise levels in decibels.
- The growth of money at a fixed rate of interest and exponential growth of population can be calculated using logarithm.
- Logarithms are also used in specific calculations where multiplications are turned into additions.
- Calculus also involves logarithmic calculations.
- Bacterial growth on a Petri dish can be found using logarithm.

### Log Table

Instead of using a calculator, a log table is used to execute larger calculations (multiplication, division, squares, and roots). When it comes to determining the value of a logarithmic function, the logarithm table is extremely useful. A log table is one of the simplest methods for obtaining the value of a given logarithmic function. The use of log table is used to find the value of a logarithmic function. The use of log tables is one of the simplest techniques to find the value of a given logarithmic function.

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