## What isthe use of logarithm???

What i mean was what is the use of logarithm in our daily lifes and can u provide me some examples with little calculations through it?? Anyway,i just registered here :)

Harris
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 Hi Harris! Logarithms were originally invented to perform calculations, since adding stuff was much easier than multiplying them. As time progressed, calculators and computers were invented, but still logarithms are useful. This doesn't stop you from still using log to calculate stuff Logarithms are also very close to the exponential function, e. This means anything concerned with rate change can be associated with logarithms, for example, chemical kinetics of reactions, population growth, radioactive decay etc. They are also be used in chemistry to find out the pH of solutions which has pH=-log[H+]. You might not use logarithms for everyday life purposes, probably like most of math. But it still is useful, depending on what you're doing.

 Quote by HarrisAz What i mean was what is the use of logarithm in our daily lifes and can u provide me some examples with little calculations through it?? Anyway,i just registered here :) Harris

Indeed: I go to the grocery store, or to the bank, or to work or to school...and never, ever have I found one single logarithm out there to help!

Anyway, if for some reason you want to solve an equation of the form $\,\,a^x=b\,\,,\,\,a,b,>0\,\,$ (for example, to know how long will

it take to your money to reach some definite ammount when invested in the bank to some fixed interest) , you can then apply logarithms

to both sides and solve for x.

DonAntonio

## What isthe use of logarithm???

Thanks guys,at least this makes any sense why i learn Loggies :D since most of the time,i learn this maths stuff but dont have the point why im learning it which what will it affect to the world something.

Anyway,Thanks guys!
Harris
 One way to define the natural logarithm is to start with the differential equation f'(x)=1/x, subject to f(1)=0, and show that there is a unique function defined on the interval (0,∞) that satisfies those conditions. We then call that function the natural logarithm. If you don't know any calculus, don't worry. The first equation above essentially says that the rate of change of the function with respect to the variable of interest is inversely proportional to the value of the variable, itself. Even without regard to any specific applications, a function with this property seems to me like a nice function to have around. It sounds like it could have a lot of applications. Defined this way, it can be proved that the natural logarithm has all of the properties you already know about logarithms. By the way, the second equation is really only there to guarantee that there only one function that satisfies the conditions. Also, there is an explicit formula for the natural logarithm; it's $f(x)=\int^{x}_{1}\frac{1}{t}dt$. If you haven't already, you will learn what these symbols mean in a beginning calculus course.
 One of the best uses of logarithms comes in differentiating a function raised to the power of a function. We can't work with something like (sinx)^3x under normal circumstances. However, if we set this equal to some number y, then we can use the extremely useful properties of logarithms to do what would otherwise be nearly impossible: (sinx)^3x = y ln[sinx)^3x]=lny 3xln[sinx] = lny Now the derivative can be taken; Using the Product rule and the chain rule, we get: 3x*(1/sinx)*cosx + ln[sinx]*3 = 1/y*(dy/dx) Thus (dy/dx) = y*(3x*(1/sinx)*cosx + ln[sinx]*3) But y = (sinx)^3x, so (dy/dx) = (sinx)^3x*(3x*(1/sinx)*cosx + ln[sinx]*3) Thank you logarithms! ^.^
 The natural logarithms is a solutions of 1 and 2. order differantial equation and we can have plotted the solution slopes. However we can plot same slope via polinoms. Now a qustion; what is the different between solution slope of differential eq. with natural log. and the polinoms without log.

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 Quote by abakal The natural logarithms is a solutions of 1 and 2. order differantial equation and we can have plotted the solution slopes. However we can plot same slope via polinoms. Now a qustion; what is the different between solution slope of differential eq. with natural log. and the polinoms without log.
I'm a bit confused by what you are asking, could you rephrase?
 The logarithm was first invented to perform hard calculations because of the identity $\textstyle \log(ab)=\log(a)+\log(b)$ Since adding large numbers is far more convenient than multiplying them by hand, logarithms were a good way to calculate, or at worst approximate this large result. However, with the development of integral calculus, it was seen that the logarithm was much more vital. Euler's Mechanica talks about the Napier constant or Euler's number, e, which is defined as $$e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}$$ or, equivalently, $$e=\sum_{k=0}^{\infty}\frac{1}{k!}$$ The constant was first discovered by Bernoulli when studying compound interests. He used the first definition. Euler was the one to prove that the first definition is equivalent to the second, and the latter converges much more rapidly. Euler also showed that $$e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ If you differentiate term by term in this series, you will see that the derivative of this function, e to the x, is the function itself. Euler named it the exponential function and it is the name we use now. The inverse of the exponential function is also important. Since exponential function satisfies the fundamental property of exponentiation, its inverse will be a logarithm, more precisely, the logarithm with base e. This logarithm is called the natural logarithm. This name follows from the simplicity of the definition of the natural logarithm: it is simply the area under the curve f(x)=1/x from 1 to n (if it was 0 to n, the area is infinite.) In other words, the area is the natural logarithm of n. The natural logarithm is denoted in two ways in mathematics. One is ln(x), following from the French equivalent of the word "natural logarithm", which is "logarithm naturalis". The other one is simply log(x), without any base notation. Since logarithms satisfy another identity, $\textstyle \log_{b}(a)=\log(a)/\log(b)$, it is possible to express any logarithm explicitly in terms of the natural logarithm. All of what we wrote can be summed up in some equations: 1) The natural logarithm is the only solution to the equation (for f(x)): $$\displaystyle \exp(f(x))=x$$ 2) The natural logarithm can be defined as follows: $$\log(n)=\int_{1}^{n}\frac{1}{x}dx$$ 3) The exponential function satisfies $$\frac{\mathrm{d}}{\mathrm{d}x}e^x=e^x$$ From these, we can easily see that 4) The natural logarithm satisfies $$\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\log(x)=\frac{1}{x}$$ The natural logarithm is involved in the solutions to many integrals. For example, we can solve the general integral $$\displaystyle \int \frac{1}{ax+b}dx$$ using the natural logarithm, using the substitution u=ax+b, du=a dx: $$\displaystyle \int \frac{1}{ax+b}dx=\displaystyle \frac{1}{a}\int \frac{1}{u}du=\frac{\log|u|}{a}+C=\frac{\log|ax+b|}{a}+C$$ The natural logarithm also arises in the concept of series. For example, consider the series $$\displaystyle \sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}$$ We can easily solve this by the Maclaurin series expansion of the natural logarithm: $$\displaystyle \log(x)=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(x-1)^k}{k}$$ Substituting x=2 simply yields the result: $$\displaystyle \log(2)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}$$ Yet another example of the natural logarithm can be the series $$\displaystyle \lim_{n\to\infty}\sum^{2n}_{k=n}\frac{1}{k}$$ which is, again, the natural logarithm of two. Logarithms are also used to describe a specific model of growth: one that has decaying rate of growth that approaches 0. Such a growth is called logarithmic growth. For example, the harmonic series grow logarithmically; because their growth at the nth term is 1/n, which approaches zero as n tends to infinity. Logarithms are involved in many of physics equations, such as Newton's Law of Cooling, which can be expressed as $$\displaystyle \frac{\mathrm{d}T}{\mathrm{d}t}=-k(T-T_a)$$ The solution of this differential equation for the time yields a logarithmic expression that involves the natural logarithm. It is worth noting that such a kind of growth (or decay) would grow (or decay) logarithmically. Finally, I will conclude with some identities involving the natural logarithm and the exponential function. $$\displaystyle e^{ix}=\cos(x) + i\sin(x)$$ $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{n} - \log(n)=\gamma$$ and is convergent. $$\int \log(x)dx=x\log(x)-x+C$$ $$\sum_{p\,prime}\log\left(\frac{p}{p-1}\right)=\infty$$ $$e^{i\pi}=-1$$
 Thanks your reply, I am wondering difference between natural log. and polinomal solutions of physical events in life which work like "cooling coffie cup". Otherwise, can we express "cooling coffie cup" event two way. and Otherwise if we draw plot of solution diff. eq. like polinom slope then solutions are the same. example; 1)dif. eq. solu=e^x x=(1:15) 2) a polinom has same solution slope with e^x ; y = 14,98x6 - 647,7x5 + 10836x4 - 87726x3 + 35118x2 - 62963x + 36923 then is this polinom become a solution of the dif. eq. like e^x
 Recognitions: Gold Member Well I'm still confused, assuming you mean polynomial, the equation you listing above seems completely arbitrary (correct me if it isn't), and just because it happens to give you the right answer doesn't mean it's the correct function. Correct me if there is more behind this than what I am seeing.
 Some real world examples of logarithms: - converting from a frequency to the position of the note on a keyboard - converting from the power output of a speaker to its decibel level - converting earthquake power to its Richter scale value - getting the pH level from a substence
 Probably I complicated topic. Yes , the equation is arbitrary. I think if we know a complex solution of dif eq. we can express easlly as a polinomal function.

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 Quote by abakal Thanks your reply, I am wondering difference between natural log. and polinomal solutions of physical events in life which work like "cooling coffie cup". Otherwise, can we express "cooling coffie cup" event two way. and Otherwise if we draw plot of solution diff. eq. like polinom slope then solutions are the same. example; 1)dif. eq. solu=e^x x=(1:15) 2) a polinom has same solution slope with e^x ; y = 14,98x6 - 647,7x5 + 10836x4 - 87726x3 + 35118x2 - 62963x + 36923 then is this polinom become a solution of the dif. eq. like e^x
I have no clue what you are talking about. First what does "x= (1:15)" mean? Second, the polynomial you give here does NOT have the same slope as e^x for any value of x. The derivative of e^x is e^x while the derivative of your polynomial is the constant -62963 and e^x is never negative.

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