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Logarithms

What is a logarithm and logarithmisation?

A logarithm is used to calculate the exponent of a power when we know the value of the power and the value of its base. A logarithm can, therefore, be most simply explained as the inverse of exponentiation.

In basic terms, the logarithm of base ‘a’ from ‘b’, we shall call ‘a’ number ‘e’ – such that ‘a’ raised to the power of ‘c ‘is equal to ‘b’.
The mathematical formula of the logarithm is as follows: log_a(b)=c <-- the given formula an be read as follows: the logarithm of the number ‘b’ with base ‘a’ is called such a number ‘c’ (i.e., such a number a raised to the power of ‘c’) that will give us the number ‘b’.
So, log_a(B)=c is nothing else except a^c=b

The three conditions necessary for a logarithm to exist:

1. a>0 i.e., the base of logarithm must always be positive and greater than zero
2. a≠1 means that the base must be different to 1
3. b>0 means that the logarithmic number must be positive

How do you calculate a logarithm?
Calculating logarithms can be quite a problem at first glance. First of all, you need to take a good look and learn the definition of a logarithm (stated above). Once this is done, we can immediately see that when solving the logarithm, we have to answer a very important question: to what power should the base (i.e., our ‘a’) be raised in order to obtain the logarithmic number (i.e., our ‘b’’)?

Examples:
- A logarithm with base 3 of the number 9 = 2
Why?
Because 3 squared gives us 9.
- A logarithm of base 3 of 81 = 4
Why?
Because 3 squared to the power of 4 gives us 81.
- A logarithm of base 2 of the number 16 = 4
Why?
Because 2 to the power of 4 gives us 16.

Basic operations on logarithms
- Addition and subtraction
When logarithms have the same base, we can freely add and subtract them using the formulas below:
log_a(b) + log_a(c) = log_a(b * c)
log_a(b) - log_a(c) = log_a(b / c)
Examples
- log_(2)2 + log_2(8)
- log_25 + log_40
- log_8(32) + log_8(2)
- log_3(36) - log_3(4)
- log_2(24) - log_2(3)
- log_100 - log_2(8)

Logarithmic equations
What is a logarithmic equation? It is an equation in which an unknown appears in the logarithmic expression or in the base of the logarithm.
To begin solving logarithmic equations, we should start with the definitions and conditions necessary for logarithms to exist. Let us remind you of them:

1. a>0 i.e., the base of the logarithm must always be a positive number and greater than zero
2. a≠1 means that the base must be different to 1
3. b>0 means that the logarithmic number must be positive

Then, to correctly solve a logarithmic equation, the equations must be brought to equality of logarithms with the same bases.

Examples of logarithmic equations
- log_2(x) = 4
- log_2(x^2+3x-8) = 1
- log_x+2(9) = 2
- 2+log_5(3x-5) = log_5(2x+23)
- log_4(x)+log_16(x)+log_64(x) = 713

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