How to express ##3^{\sqrt(2)}## in terms of natural logarithms

  • #1
MevsEinstein
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So in my Calculus book, it asked a question in its Transcendental Functions chapter. It wanted me to express ##3^{\sqrt{2}}## in terms of natural logarithms I have no idea how to solve this. All I know is that ##3^{\sqrt{2}} = e^{\sqrt{2}\ln{3}}## but that's not completely in natural logarithm form.
 
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  • #3
fresh_42
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Summary:: What the title says

So in my Calculus book, it asked a question in its Transcendental Functions chapter. It wanted me to express ##3^{\sqrt{2}}## in terms of natural logarithms I have no idea how to solve this. All I know is that ##3^{\sqrt{2}} = e^{2\ln{3}}## but that's not completely in natural logarithm form.
You should correct your mistake, then write ##x=3^{\sqrt{2}}## and transform it via logarithm and exponentiation.
 
  • #4
mathman
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Maybe ##ln(3^{\sqrt{2}})=\sqrt{2}ln(3)?##
 
  • #5
MevsEinstein
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Maybe ##ln(3^{\sqrt{2}})=\sqrt{2}ln(3)?##
The question was to represent ##3^\sqrt{2}## as natural logarithms. ##\sqrt{2} \ln{3}## won't be the answer
 
  • #6
fresh_42
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The question was to represent ##3^\sqrt{2}## as natural logarithms. ##\sqrt{2} \ln{3}## won't be the answer
Again. Set ##x=3^{\sqrt{2}}## and rewrite it as ##x=e^{\ln x}## with appropriate changes on the right.
 
  • #7
nuuskur
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## \ln \exp \left (3^{\sqrt{2}}\right) ## seems to do the trick
 
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  • #8
Office_Shredder
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I think this question is underspecified. Do you have some other examples with correct solutions so we can try to figure out what they're actually looking for?
 
  • #9
MevsEinstein
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I think this question is underspecified. Do you have some other examples with correct solutions so we can try to figure out what they're actually looking for?
The book didn't have any other examples.
 
  • #10
Office_Shredder
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: maybe you're supposed to do the same thing to the ##\sqrt{2}## that you did to the 3?
 
  • #11
fresh_42
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Or simply: ##\log_3 x= \log_3 \left(3^{\sqrt{2}}\right)=\sqrt{2}##.
 

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