How do you simplify 9e^(2y) = x^2?

Question

could someone please remind me what the rules of natural logs are?

Answer 1

9e^(2y) = x^2
assuming that what you want is to write y as a function of x:
e^{2y} = x^2 /9
2y = ln( x^2 / 9)
y = ln( x^2 / 9) /2
one can simplify further:
y = ln( x^2 / 9) /2 = ( 2ln(x) - ln(9) ) /2 = ln(x) - 2ln(3)/2 = ln(x) -ln(3)
= ln(x/3) .

Answer 2

When you ln-ize you should remember that what you ln-ize should always be a positive quantity
ln(ab)=lna+lnb, ln(a/b)=lna-lnb, ln(a^b)=blna and these identities are valid for any type of logarithms.
For natural logarithms in particular, lne=1, so,
supposing that x^2 >0 (and so is the other part of the equation), you have
9e^(2y) = x^2, ln[9e^(2y)] = ln(x^2), or
ln9+ln[e^(2y)] = ln(x^2), 2ylne=2lnx-ln9, 2y=2lnx-ln9,
y=lnx-(ln9)/2=lnx-(1/2)ln9
=lnx-ln[9^(1/2)]=ln(x/3)

Answer 3

9e^(2y) is (3e^y)^2 since 9=3^2 and e^(2y) = (e^y)^2

so (3e^y)^2 = x^2 which gives taking root both side

x = (+/-) 3e^y

Answer 4

x^2 = 9e^2y
x = (9e^2y)^0.5 = 3e^y

Answer 5

9e^(2y) = x²
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x =e^y => y = ln(x)
ln(e^y) = y
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9e^(2y) = x²
=> e^(2y) = x²/9
=> e^(2y) = x²/9
=> ln(e^(2y)) = ln(x²/9)
=> 2y = ln(x²/9)

=> y = ln(x²/9)/2

Answer 6

9e^(2y) = x^2
9ln(x^2) = 2y
18ln(x) = 2y
9ln(x) = y

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Let ln(x) = y

Then,

e^y = x