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QUESTION

How do you simplify e^-lnx?

e^(-ln(x))" " =" " 1/x

color(brown)("Total rewrite as changed my mind about pressentation.")

color(blue)("Preamble:")

Consider the generic case of " "log_10(a)=b

Another way of writing this is 10^b=a

Suppose a=10 ->log_10(10)=b

=>10^b=10 => b=1

So color(red)(log_a(a)=1 larr" important example")

We are going to use this principle. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Write " "e^(-ln(x))" " as " "1/(e^(ln(x))

Let y=e^(ln(x)) =>" "1/y=1/(e^(ln(x)) ..................Equation(1)

....................................................................................... Consider just the denominators and take logs of both sides

y=e^(ln(x))" " ->" "ln(y)=ln(e^(ln(x)))

But for generic case ln(s^t) -> tln(s)

color(green)(=>ln(y)=ln(x)ln(e))

But log_e(e)" "->" "ln(e)=1 color(red)(larr" from important example")

color(green)(=>ln(y)=ln(x)xx1)

Thus y=x .....................................................................................

So Equation(1) becomes

1/y" "=" "1/(e^(ln(x)))" "=" "1/x

Thus e^(-ln(x)) = 1/x

~~~~~~~~~~~~~~~~~~~~~~~~~~~~ color(blue)("Footnote")

In conclusion the general rule applies: " "a^(log_a(x))=x