Question

give the proof of exponential theoram ​

Answer 1

Let ax=y.

Let ax=y.xlna=lny

Let ax=y.xlna=lnyexlna=y

Let ax=y.xlna=lnyexlna=y∴ax=exlna

Let ax=y.xlna=lnyexlna=y∴ax=exlnaax=(ex)lna

Let ax=y.xlna=lnyexlna=y∴ax=exlnaax=(ex)lnaaxlna=ex

Let ax=y.xlna=lnyexlna=y∴ax=exlnaax=(ex)lnaaxlna=exSubstituting axlna=ex into the Maclaurin expansion for ex should give you the same expansion as your original one.

Let ax=y.xlna=lnyexlna=y∴ax=exlnaax=(ex)lnaaxlna=exSubstituting axlna=ex into the Maclaurin expansion for ex should give you the same expansion as your original one.(Note: this may be wrong so please pardon me. I am also in the 10th grade and I am supposed to learn this after 2 years).

Answer 2

Answer:

By Taylor series we have:

f(x)=∑n=0∞f(n)(0)n!xn

with f(x)=ax=exlna see that

f′(x)=lna exlna=lna⋅f(x), f(2)(x)=(lna)2f(x)⋯⋯,f(n)(x)=(lna)nf(x)

that is

f(0)=1, f′(0)=lna, f(2)(0)=(lna)2⋯⋯,f(n)(0)=(lna)n