Why b^[log (base b) a ] is = a ?

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The equation b^[log (base b) a] = a is defined as b^[log_b a] ≡ a, meaning it is true by definition rather than something that can be proven. The logarithm definition states that log_b x = y if and only if b^y = x, establishing the foundational relationship. Since definitions serve as the basis for mathematical concepts, they cannot be proven like theorems. An analogy is drawn to the relationship between inches and feet, which also cannot be proven but is accepted as a definition. Understanding this distinction is crucial in mathematics.
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I was wondering if anyone could show me why b^[log (base b) a ] is = a ?
 
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The equation

b^{\log_b a} = a

is better written as

b^{\log_b a} \equiv a

In other words, the equality is true by definition.
 


D H said:
The equation

b^{\log_b a} = a

is better written as

b^{\log_b a} \equiv a

In other words, the equality is true by definition.

is it possible to prove it ? like simplify it then showed that it is equal to a ?
 


It is not possible to prove a definition, and the standard definition of \log_b x is

\log_b x = y \,\Leftrightarrow \, b^y = x
 


garyljc said:
is it possible to prove it ? like simplify it then showed that it is equal to a ?

You don't prove definitions. They are the foundational building blocks that provide meanings. As an analogy, a dictionary is more or less a long list of word and definition pairs.

A simpler example than the one you posted is: 12 inches = 1 foot. This equation defines the relationship between inches and feet and is not something that is proved.
 

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