- #1
Sturk200
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How is it true that:
Log[L+(Z^2+L^2)^(1/2)] - Log[-L+(Z^2+L^2)^(1/2)] = 2{Log[L+(Z^2+L^2)^(1/2)] - Log[Z]}
?
Log[L+(Z^2+L^2)^(1/2)] - Log[-L+(Z^2+L^2)^(1/2)] = 2{Log[L+(Z^2+L^2)^(1/2)] - Log[Z]}
?
DrClaude said:Is this homework?
The equality is not very complicated to check. Use the properties of logarithms to get to the form log(x) = log(y), and then check if x = y.
Is your equation:Sturk200 said:How is it true that:
Log[L+(Z^2+L^2)^(1/2)] - Log[-L+(Z^2+L^2)^(1/2)] = 2{Log[L+(Z^2+L^2)^(1/2)] - Log[Z]}
?
SteamKing said:Is your equation:
##log\; [L + \sqrt{(Z^2 + L^2)}] -log\; [-L + \sqrt{(Z^2 + L^2)}]=2log\; [L + \sqrt{(Z^2 + L^2)}] - log (Z)## ?
The Laws of Logarithms are:
##log\; (a) - log\; (b) = log\;(\frac{a}{b}) ##
##log\; (a+b) - log\; (a-b) = log\;(\frac{a+b}{a-b}) ##
##log\;(a^b) = b\;log\;(a)##
##log\;[(a+b)^c]=c\;log\;(a+b)##
If you start from the left side, simply multiply by ##(L+\sqrt{Z^2+L^2})/(L+\sqrt{Z^2+L^2})##. This is the same trick as when we multiply by conjugates of complex numbers to get rid of a term (a+ib) in the denominator.Sturk200 said:Thanks for your reply. Yes, I understand this much. So the problem becomes showing that
(L + Sqrt[Z^2 + L^2])/(-L + Sqrt[Z^2 + L^2]) = [(L + Sqrt[Z^2 + L^2])/Z]^2
Maybe this is me being dumb, but I don't know how to get from the left side to the right side.
nrqed said:If you start from the left side, simply multiply by ##(L+\sqrt{Z^2+L^2})/(L+\sqrt{Z^2+L^2})##. This is the same trick as when we multiply by conjugates of complex numbers to get rid of a term (a+ib) in the denominator.
Natural log, written as ln, is a mathematical function that is the inverse of the exponential function. It is used to determine the amount of time needed for a quantity to reach a certain value based on its growth rate.
To manipulate natural log, you can use various mathematical properties such as the product rule, quotient rule, and chain rule. These rules allow you to simplify and solve complex equations involving natural log.
The main purpose of manipulating natural log is to solve equations that involve exponential growth or decay. It is also used in various fields such as finance, physics, and biology to model and analyze natural phenomena.
No, natural log cannot be used for negative numbers. The natural log function is only defined for positive numbers, as the logarithm of a negative number is undefined.
Natural log and the natural exponential function, written as e^x, are inverse functions of each other. This means that if you take the natural log of a number and then raise it to the power of e, you will get the original number back. This relationship is fundamental in many mathematical and scientific applications.