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

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• MevsEinstein
In summary, the conversation discusses the process of representing ##3^{\sqrt{2}}## in terms of natural logarithms. The individual does not know how to solve this but does know that ##3^{\sqrt{2}} = e^{2\ln{3}}##. However, this is not completely in natural logarithm form. The conversation then explores different ways to approach the problem, including setting ##x=3^{\sqrt{2}}## and rewriting it as ##x=e^{\ln x}##. It is noted that the question may be underspecified and the individual does not have any other examples from the book to reference.
MevsEinstein
TL;DR 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^{\sqrt{2}\ln{3}}## but that's not completely in natural logarithm form.

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

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.
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)?##

mathman said:
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

MevsEinstein said:
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
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?

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?
The book didn't have any other examples.

: maybe you're supposed to do the same thing to the ##\sqrt{2}## that you did to the 3?

Or simply: ##\log_3 x= \log_3 \left(3^{\sqrt{2}}\right)=\sqrt{2}##.

## 1. What is the formula for converting ##3^{\sqrt(2)}## to natural logarithms?

The formula for converting a number raised to a power to natural logarithms is: ln(ab) = b * ln(a). In this case, a = 3 and b = √2.

## 2. How do I solve for the value of ##3^{\sqrt(2)}## in terms of natural logarithms?

Using the formula from the previous question, we can solve for the value of ##3^{\sqrt(2)}## in terms of natural logarithms. First, we rewrite the expression as ##e^{ln(3^{\sqrt(2)})}##. Then, we can use the rule for logarithms to rewrite it as ##e^{ln(3) * √2}##. Finally, we can use a calculator or the value of ln(3) to calculate the approximate value of ##3^{\sqrt(2)}## in terms of natural logarithms.

## 3. What is the significance of using natural logarithms to express ##3^{\sqrt(2)}##?

Natural logarithms, represented by ln(), are logarithms with a base of e (Euler's number). They are often used in scientific and mathematical calculations because they have many useful properties and can simplify complex expressions. In this case, using natural logarithms allows us to express ##3^{\sqrt(2)}## in a simpler and more compact form.

## 4. Can I express ##3^{\sqrt(2)}## in terms of other logarithms?

Yes, you can express ##3^{\sqrt(2)}## in terms of other logarithms, such as base 10 or base 2. However, using natural logarithms is often preferred due to their useful properties and the fact that e is often used as the base in scientific and mathematical calculations.

## 5. How can I use the value of ##3^{\sqrt(2)}## in terms of natural logarithms in practical applications?

The value of ##3^{\sqrt(2)}## in terms of natural logarithms can be used in various scientific and mathematical calculations, such as in exponential growth and decay problems, population growth models, and financial calculations. It can also be used to simplify complex expressions and equations in these fields.

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