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

**a ^{n} = 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:

**log _{a}b = n → a^{n }= 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**

**a**, every number raised to zero is equal to one. Whenever a ≠ 0. Example:^{0 }= 1

5^{0 }= 1 ; 80^{0 }= 1; π^{0 }= 1; √230^{0 }= 1

**a**, every number raised to one is equal to the same number. Example:^{1 }= to

7^{1 }= 7 ; 56^{1 }= 56; (∛4)^{1 }= ∛4; (1.65)^{1 }= 1,65

**a**, if the product of powers have the same base, the base is conserved and its exponents are added. Example:^{n}a^{m }= to^{n + m}

3^{8}·3^{2 }= 3^{10} ; 4^{15}·4^{-3 }= 4^{12}

**a**, if we have division of powers, the base is conserved and its exponents are subtracted. Example:^{n}/to^{m}= to^{nm}oa^{n}÷ a^{m }= to^{nm}

2^{5} ÷ 2^{12 }= 2^{-7} ; 8^{-3}/8^{6} = 8^{-9} ; 6^{4 }÷ 6^{-4 }= 6^{8}

**(to**if we have a power raised to an exponent, the base is conserved and the exponents are multiplied. Example:^{n})^{m }= to^{n}^{m},

(5^{5} )^{5 }= 5^{25} ; (9^{2})^{-3 }= 9^{-6} ; (-4^{3})^{4 }= (-4)^{12}

**(ab)**If we have a power where its base is a product, the exponent will affect each of the components of the base. Example:^{n }= to^{n}B^{n}

(6 · 8)^{3} = 6^{3}8^{3} ; (-5 · 2)^{6 }= (-5)^{6}·two^{6} ; (9 · 3)^{-4} = 9^{-4}·3^{-4}

**(a / b)**= to^{n }= to^{n}/ b^{n}or (a ÷ b)^{n }^{n }÷ b^{n}, if we have a power where its base is a division, the exponent will affect the components of the base. Example:

(5/2)^{8 }= 5^{8}/2^{8} ; [6 ÷ (-8)]^{-4 }= 6^{-4} ÷ (-8)^{-4}

**a**, 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:^{-n}= 1 / a^{n}

7^{-6 }= 1/7^{6}

**1 / a**, the same thing happens with property (8), only in the opposite way. Example:^{-n}= to^{n}

1/7^{-3} =7^{3}

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

(9/7)^{-2} = (7/9)^{2}

**Properties of logarithms**

**log**, the logarithm of one to any base is zero. Example:_{a}1 = 0

log_{15} 1 = 0; log_{230} 1 = 0

**log**, the logarithm of a number equal to the number of the base is one. Example:_{a}a = 1

log_{2} 2 = 1; log_{80} 80 = 1

**log**, the logarithm of a number equal to the number of the base raised to an exponent, will be equal to the exponent. Example:_{a }a^{n}= n

log_{12} 12^{3 }= 3

**log**, 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:_{a}(nm) = log_{a}n + log_{a}m

log_{2} (3 · 5) = log_{2} 3 + log_{2} 5

**log**, the logarithm of a power is equal to the product of the exponent, by the logarithm of the base. Example:_{a}x^{n}= n log_{a}x

log_{3} 8^{2 }= 2 · log_{3} 8

**log**, this is a particular case of the power, where the index of the root stops to divide the logarithm of the base. Example:_{a}^{n}√x = (log_{a}x) / n

log_{4} √3 = (log_{4} 3) / 2

**log**, 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:_{a}(n / m) = log_{a}n - log_{a}m

log_{3} (2/27) = log_{3} 27 - log_{3} 2