The discussion focuses on expressing the mathematical expression ##3^{\sqrt{2}}## in terms of natural logarithms. Participants confirm that the expression can be represented as ##e^{\sqrt{2} \ln{3}}##, but debate whether this is the desired form. The transformation of the expression using logarithmic properties is emphasized, particularly the relationship ##\ln(3^{\sqrt{2}}) = \sqrt{2} \ln(3)##. The conversation highlights the need for clarity in the problem statement, as participants question the sufficiency of the provided information.
PREREQUISITESStudents studying calculus, particularly those focusing on transcendental functions, as well as educators seeking to clarify logarithmic transformations in mathematical expressions.
Are you sure?MevsEinstein said:All I know is that ##3^{sqrt{2}} = e^{2*ln{3}}##
Perhaps that form is what they want?MevsEinstein said: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.MevsEinstein said: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.
The question was to represent ##3^\sqrt{2}## as natural logarithms. ##\sqrt{2} \ln{3}## won't be the answermathman said:Maybe ##ln(3^{\sqrt{2}})=\sqrt{2}ln(3)?##
Again. Set ##x=3^{\sqrt{2}}## and rewrite it as ##x=e^{\ln x}## with appropriate changes on the right.MevsEinstein said:The question was to represent ##3^\sqrt{2}## as natural logarithms. ##\sqrt{2} \ln{3}## won't be the answer
The book didn't have any other examples.Office_Shredder said: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?