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Niaboc67
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Hello, I am beginning to learn precalculus. I understand that there are times where you can change logarithms to exponential expressions. So how are they different and similar and why are they interchangeable?
Niaboc67 said:Forgive me but, x, m and n represent what exactly?
It should be ##m=\log_x(n)## or m=log_x(n)Niaboc67 said:m=logx^(n)?
No. I use brackets for my own convenience. I also use brackets in trigonometric functions. It makes things neat.Niaboc67 said:What is the purpose of the parentheses? does that multiply out something?
Hello and Welcome!Niaboc67 said:So how are they different and similar and why are they interchangeable?
sudhirking said:If you were to square the square root you would have gotten back to the original value. Or even if you square root the square they would have cancelling actions.
(sqrt(x))^2=sqrt (x^2)=x
You would agree that these two things are equivalent:
x^2=y and y= sqrt(x).
micromass said:I disagree with this. Square root and squares are not inverses and do not always cancel each other out. The right relation is
[tex]\sqrt{x^2} = |x|[/tex]
Under the special condition that ##x\geq 0##, then ##x^2 = y## and ##y= \sqrt{x}## are equivalent. Not in general.
Logarithms and exponentials are mathematical operations that are essentially inverse of each other. While exponentials are used to determine the power to which a base number is raised, logarithms are used to determine the exponent required to produce a given number.
The main properties of logarithms and exponentials include the commutative property, which states that changing the order of the numbers in an operation does not change the result; the associative property, which states that grouping numbers in different ways does not change the result; and the distributive property, which states that multiplying a number by a sum is the same as multiplying the number by each addend separately and then adding the products.
Logarithms and exponentials are used in various fields such as finance, physics, biology, and engineering. In finance, they are used to calculate compound interest and growth rates. In physics, they are used to model exponential decay and growth. In biology, they are used to measure the pH scale and the decibel scale. In engineering, they are used in signal processing and control systems.
The relationship between logarithms and exponentials can be expressed as: logb(x) = y if and only if by = x. In other words, logarithms and exponentials are inverse functions of each other, meaning that taking the logarithm of a number is the inverse of raising that number to a power.
Natural logarithms use the base e, which is an irrational number approximately equal to 2.718, while common logarithms use the base 10. Natural logarithms are often used in calculus and mathematical analysis, while common logarithms are used in everyday calculations and in various fields such as finance and engineering.