It is because f(x) = e^x and g(x) = ln x are inverse functions. This means that f(g(x)) = x
f(g(x)) = e^(lnx) = x.
Or you can think about how ln is defined.
ln x is defined to be the value y such that e^y = x
So if x = 7 and ln x = y
then substitution in e^y = x gives you.
e^(ln7) = 7
E Ln 7
Because if you take the ln of both sides, you get an equality.
e^ln7 = 7
LN BOTH SIDES:
ln(e^ln7) = ln7
when you take the natural log of e to something, the e goes away, so
ln7 = ln7
Since this equation is true, the one before also has to be true, because we just natural logged both sides–that doesn’t change the equality.
it doesn’t it equals 1.945910
ohh e^ln7 you mean?
haha because if e has a power of ln then it cancles the e such as e^ln(1)=1 e^ln59289455023 = 59289455023
The function ln(x) is the inverse of e^x and defined such that e^(ln(x)) = x.
The definition of logarithm tells us:
ln(7) = a ⇐⇒ ℯª = 7 ← notice that a = ln(7)
ℯʵⁿ⁽⁷⁾ = 7 ← by substitution
ln is base e.
The e’s cancel out
so all you’re left with 7=7.
Have a nice day
Source(s): Grade 10 math