# What isthe use of logarithm?

1. May 13, 2012

### HarrisAz

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

2. May 13, 2012

### Infinitum

Re: What isthe use of logarithm???

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.

3. May 13, 2012

### DonAntonio

Re: What isthe use of logarithm???

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

4. May 13, 2012

### HarrisAz

Re: 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

5. May 21, 2012

### laughingebony

Re: What isthe use of logarithm???

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.

6. May 25, 2012

Re: What isthe use of logarithm???

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! ^.^

7. Jun 7, 2012

### abakal

Re: What isthe use of logarithm???

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.

8. Jun 7, 2012

### Vorde

Re: What isthe use of logarithm???

I'm a bit confused by what you are asking, could you rephrase?

9. Jun 7, 2012

### Millennial

Re: What isthe use of logarithm???

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.
$$\int \log(x)dx=x\log(x)-x+C$$
$$\sum_{p\,prime}\log\left(\frac{p}{p-1}\right)=\infty$$
$$e^{i\pi}=-1$$

Last edited: Jun 7, 2012
10. Jun 7, 2012

### abakal

Re: What isthe use of logarithm???

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

11. Jun 7, 2012

### Vorde

Re: What isthe use of logarithm???

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.

12. Jun 7, 2012

Re: What is the use of logarithm???

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

13. Jun 8, 2012

### abakal

Re: What isthe use of logarithm???

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.

14. Jun 8, 2012

### HallsofIvy

Re: What isthe use of logarithm???

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.

15. Jun 8, 2012

### HallsofIvy

Re: What isthe use of logarithm???

Now, we can't. We can approximate an exponential by a polynomial but that is true of many sets of functions, not just polynomials.

16. Jun 8, 2012

### HallsofIvy

Re: What isthe use of logarithm???

The real difficulty with understanding what you are asking is that you are asking "why can we do this" and "why can we do that" when, in fact, we can't do either. Perhaps you are confusing "expressing by" with "approximating".

17. Jun 8, 2012

### Xitami

Re: What isthe use of logarithm???

18. Jun 10, 2012

### NegativeDept

Re: What is the use of logarithm???

These are good examples, IMO. Here are a few more:
- Calculating compound interest and/or time value of money
- Internet security (RSA decryption is a discrete-logarithm problem)
- Detecting fraud in accounting and scientific publication (see Benford's Law)
- Figuring out the half-life of radioactive stuff
- Almost anything that involves entropy or thermodynamics

We could debate forever about how "real-world" these examples are. For example, knowledge of thermodynamics is necessary to design an internal-combustion engine. But you don't need $S = k_B \log(\Omega)$ to drive a car.

19. Jun 12, 2012

Re: What isthe use of logarithm???

Some more-
time in ms for insect to get within certain distance of landing point
number of minutes before coffee gets within certain distance from room temperature
number of hours that a body has been dead based on its temperature
time taken to search a database of a particular size (often its a log time)
number of digits in a large number
number of generations needed to breed a certain number of mice

In other words, a log counts how many times you need to scale something by a value.