Proof of the fact that \(\displaystyle{\ln{{\left({a}\right)}}}={f}'{\left({0}\right)}\) for

Zoie Phillips

Zoie Phillips

Answered question

2022-04-02

Proof of the fact that ln(a)=f(0) for f(x)=ax?

Answer & Explanation

glikozyd3s68

glikozyd3s68

Beginner2022-04-03Added 16 answers

Hint:
Use the definition
f(0)=limh0f(0+h)f(0)h
=limh0ah1h
limh0elnah1h
=limh0ehlna1h
=limh01+hlna+1h=lna
Mikaela Winters

Mikaela Winters

Beginner2022-04-04Added 14 answers

First use the commonly known limit "limeh1h", that is:
1=limx0ex1x
=limlog(a)x0elog(a)x1log(a)x
=limx01log(a)ax1x
and so f(0)=log(a) by the definition of the derivative.

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