How does [itex]a^{x}=e^{log_{e} a^{x}}[/itex]

[Lebombo]

What are the concepts/principles that let a^{x}=e^{log_{e} a^{x}}

 

[DeIdeal]

What's your definition of the logarithm & exponential function? Using the most common ones, e^{\ln(x)}=x should be fairly obvious.

 

[symbolipoint]

The logarithm function and the exponential function are inverse functions. Look carefully at your equation and see that the exponential function uses the logarithm function as its input, and continue to see that the input of that logarithm function is a^x. This means that the output of the composition is a^x.

 

[lurflurf]

I normally define a^x by
$$a^x=e^{\log (a) \, x}$$
whatever definition you adopt that equation should be obvious from
$$a^x a^y=a^{x+y}$$

 

[Lebombo]

So in the same way that the inverse of the exponent can be represented like so...

[2^{3} = 8] = [log_{2}8 = 3]

...we can arrange the inverse of this exponent in question like so:

[e ^{log_{e} a^{x}} = a^{x}] => [log_{e} a^{x} = log_{e}a^{x}](Color coordinated so that, if correct, it can be determined to be correct for the right reason - that is, arranged correctly.)

 

[lurflurf]

yes
another way to think about it is to think of a^x as one number say b=a^x then
$$[b=e^{\log_e (b)}] = [\log_e (b)=\log_e (b) ] $$

 

[Lebombo]

LurfLurf, symbolipoint, and DeIdeal,

Thank you very much for the replies.