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?
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.
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.
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.
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.
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.