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

MevsEinstein
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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^{\sqrt{2}\ln{3}}## but that's not completely in natural logarithm form.

Last edited:

Homework Helper
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All I know is that ##3^{sqrt{2}} = e^{2*ln{3}}##
Are you sure?
but that's not completely in natural logarithm form.
Perhaps that form is what they want?

Mentor
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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.

Maybe ##ln(3^{\sqrt{2}})=\sqrt{2}ln(3)?##

MevsEinstein
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

Mentor
2022 Award
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.

• Klystron
## \ln \exp \left (3^{\sqrt{2}}\right) ## seems to do the trick

• • mfb and PeroK
Staff Emeritus
Gold Member
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?

MevsEinstein
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.

Staff Emeritus