What is logarithm?

Logarithm is defined as an operation contrary to potentiation or exponential.

In potentiation, we know the base and the exponent and we want to calculate a power. In the logarithm, we know the base and the power and we want to know the value of the exponent.

So, realize that logarithm is not the radiciation, since in the latter we look for the base value given the power.

Example: What should the value of the exponent x be for

$\dpi{120} \mathrm{5^x = 25}?$

We know that $\dpi{120} 5^2 = 25$ , then the exponent x must be equal to 2.

So we can say that the logarithm of 25 in base 5 is equal to 2:

$\dpi{120} \mathrm{log\, _5\, 25} = 2$

See below for a formal definition of logarithm.

Definition of logarithm:

Given two positive numbers, The and B, with $\dpi{120} \mathrm{a\neq 1}$ , we say that the logarithm of B at the base The is equal number x if, and only if, The raised to x it's the same as B, that is:

$\dpi{150} \mathbf{\log_a b = x \Leftrightarrow a^x = b}$

On what:

The: base
B: logarithm
x: logarithm

Example: Calculate the value of $\dpi{120} \mathrm{x}$ in each case.

The) $\dpi{120} \mathrm{\log_9 81 = x}$

By definition, we have to:

$\dpi{120} \mathrm{9^x = 81}$

Like $\dpi{120} 9^2 = 81$ , then, $\dpi{120} \mathrm{x= 2}$ . Thus:

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$\dpi{120} \mathrm{\log_9 81 = 2}$

B) $\dpi{120} \mathrm{\log_2 8 = x}$

By definition, we have to:

$\dpi{120} \mathrm{2^x = 8}$

Like $\dpi{120} 2^3 = 8$ , then, $\dpi{120} \mathrm{x= 3}$ . Thus:

$\dpi{120} \mathrm{\log_2 8 = 3}$

Logarithm Properties

From the definition of logarithms, we have the following immediate results:

1) $\dpi{120} \mathrm{log_a1 = 0}$

2) $\dpi{120} \mathrm{log_aa = 1}$

3) $\dpi{120} \mathrm{log_aa^c = c}$

4) b = c ⇒ $\dpi{120} \mathrm{log_ab = log_ac}$

5) $\dpi{120} \mathrm{a^{log_ab} = b}$

And the logarithm properties they are:

1) $\dpi{120} \mathrm{log_a (b\cdot c) = log_ab + log_ac}$

2) $\dpi{120} \mathrm{log_a\bigg(\frac{b}{c} \bigg) = log_ab - log_ac}$

3) $\dpi{120} \mathrm{log_ab^c = c\cdot log_ab}$

4) $\dpi{120} \mathrm{log_ab = \frac{log_cb}{log_ca}}$

You may also be interested:

Logarithm Exercise List
List of potentiation exercises
Radiation exercises

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