# Properties of the potentiation and logarithm

In math when we talk about power, we refer to an abbreviated multiplication and it is expressed in the following way:

an = a · a · a · a · a… · a = b

We can say, that is the multiplication of a, n times. Where, a represents the base, n is the exponent and together they are a power.

The logarithm it is intimately linked to the power, provided that a> 0 and ≠ 1. Let us observe the two expressions:

logab = n → an = b

That is, the logarithm to the base of b is the exponent to which we must raise a so that the result is b. Where, a is the base, b is the argument of the logarithm or power and n is the exponent on the power or logarithm.

## Exponentiation properties

• a0 = 1, every number raised to zero is equal to one. Whenever a ≠ 0. Example:

50 = 1    ;     800 = 1; π0 = 1; √2300 = 1

• a1 = to, every number raised to one is equal to the same number. Example:

71 = 7     ;    561 = 56; (∛4)1 = ∛4; (1.65)1 = 1,65

• anam = ton + m, if the product of powers have the same base, the base is conserved and its exponents are added. Example:

38·32 = 310      ;      415·4-3 = 412

• an/tom = tonm oan÷ am = tonm, if we have division of powers, the base is conserved and its exponents are subtracted. Example:

25 ÷ 212 = 2-7      ;     8-3/86 = 8-9       ;      64 ÷ 6-4 = 68

• (ton )m = tonm, if we have a power raised to an exponent, the base is conserved and the exponents are multiplied. Example:

(55 )5 = 525      ;        (92)-3 = 9-6       ;    (-43)4 = (-4)12

• (ab)n = tonBnIf we have a power where its base is a product, the exponent will affect each of the components of the base. Example:

(6 · 8)3 = 6383         ; (-5 · 2)6 = (-5)6·two6       ; (9 · 3)-4 = 9-4·3-4

• (a / b)n = ton/ bn or (a ÷ b)n = ton ÷ bn, if we have a power where its base is a division, the exponent will affect the components of the base. Example:

(5/2)8 = 58/28         ; [6 ÷ (-8)]-4 = 6-4 ÷ (-8)-4

• a-n = 1 / an, if we have a power with negative exponent, that is equal to a fraction, with numerator one and denominator with the same base but the positive exponent. Example:

7-6 = 1/76

• 1 / a-n = ton, the same thing happens with property (8), only in the opposite way. Example:

1/7-3 =73

• (a / b)-n = (b / a)n, if we have a fraction raised to a negative exponent, this will be equal to the inverted fraction but with the positive exponent. Example:

(9/7)-2 = (7/9)2

## Properties of logarithms

• loga1 = 0, the logarithm of one to any base is zero. Example:

log15 1 = 0; log230 1 = 0

• loga a = 1, the logarithm of a number equal to the number of the base is one. Example:

log2 2 = 1; log80 80 = 1

• loga an = n, the logarithm of a number equal to the number of the base raised to an exponent, will be equal to the exponent. Example:

log12 123 = 3

• loga (nm) = loga n + loga m, the logarithm of base a of a product, will be equal to the sum of the logarithms of base a of each of the factors. Example:

log2 (3 · 5) = log2 3 + log2 5

• loga xn = n loga x, the logarithm of a power is equal to the product of the exponent, by the logarithm of the base. Example:

log3 82 = 2 · log3 8

• loga n√x = (loga x) / n, this is a particular case of the power, where the index of the root stops to divide the logarithm of the base. Example:

log4 √3 = (log4 3) / 2

• loga (n / m) = loga n - loga m, the logarithm of base a of a division or fraction, will be equal to the difference or subtraction of the logarithms of each one of the factors. Example:

log3 (2/27) = log3 27 - log3 2