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

[garyljc]

I was wondering if anyone could show me why b^[log (base b) a ] is = a ?

 

[D H]

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.

 

[garyljc]

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 ?

 

[D H]

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

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

 

[Mark44]

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.