give the proof of exponential theoram
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give the proof of exponential theoram
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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:
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