Why is a=e^lna ??????

Is it because:

lna=log(base e)a

log(base e)=lna

then

lna=lna

Thank You.

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It looks to me like you are working under a misconception.
So, with that in mind consider this ... logarithms are exponents.

For example,
i. log₅N is the exponent of 5 that produces N ... i.e. 5^log₅N = N
ii. log₃N is the exponent of 3 that produces N ... i.e. 3^log₃N = N
iii. log₈N is the exponent of 8 that produces N ... i.e. 8^log₈N = N

In general,
logₓN is the exponent of x that produces N ... i.e. x^logₓN = N

So,
logₑa  is the exponent of e that produces 'a' ... i.e. e^logₑa = a
           ... which is the same as saying ...
ln(a)  is the exponent of e that produces 'a' ... i.e. e^ln(a) = a

Hope that helps clear things up a bit.
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Note:
There are three commonly used logarithms which
have been assigned their own notation.
           i. Natural Logarithms ... base e logarithms
              Natural Logarithms are denoted by ln
              so that ln(m) is the natural logarithm of m.
              Note that ln(m) is 'shorthand' for logₑ(m).

         ii. Common Logarithms ... base 10 logarithms.
             Common Logarithms are denoted by log
             so that log(m) is the common logarithm of m.
             Note that log(m) is 'shorthand' for log₁₀(m).

        iii. Base 2 Logarithms
           Base 2 Logarithms are denoted by lg by some
           (note that this is an accepted notation for base 2 logarithms,
           but, not everyone uses it).
           So, lg(m) denotes the logarithm base 2 of m.
           Note that lg(m) is 'shorthand' for log₂(m).



Have a good one!
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Because BY DEFINITION, the exponential function is the INVERSE of the natural logarithm.

Putting a value through a function, and then through that function's inverse should reveal the ORIGINAL VALUE (provided it is within the proper domain).

Your proof is more complicated than it needs to be. There is no reason to talk about log base ten, when we are talking about natural logarithm and natural exponential.

When f and g are inverses, f(g(x)) = g(f(x)) = x.

LN and e are inverses.