Real-Life Applications of Logarithms

[cracker]

Ok I have taken Geometry, Algebra one and two and have 2 more days of pre cal left and I have been worken with logs and stuff like solveing, simplyfing and factoring them but I never found out what they are for... I know how to solve the problems but what do they apply to in real life?

 

[berkeman]

Logarithms are extremely useful for dealing with numbers that have a wide range. Like radio signals, for example. When you transmit hundreds of kW (kiloWatts) of power, the electric field strength by the transmitting antenna is very large. But many miles away, the signal strength has dropped to a very tiny value. The best way to deal with such wide-ranging numbers is to talk in logs, or powers of 10. The unit of a dB (decibel) is based on logs:

V[dB] = 20 * log (V)

or for dealing with power:

P[dBm] = 10 * log (P/1mW) (for dB above a milliWatt).

So every power of 10 decrease in the amplitude of a voltage or field strength, you get a -20dB decrease in the value. And for every power of 10 increase in the power, you get a +10dB increase. It's a lot easier many times to deal with dB instead of powers of 10.

And different bases are used for specialized logs, like base e for natural logs, or ln.

 

[HallsofIvy]

The most important mathematical application of logarithms is to solving equations involving exponentials. For b> 0, log_b(x) is defined to be the inverse function to f(x)=b^x. Because of that, $log_b(b^x)= x$ and $b^{log_b(x)}= x$. It is also helpful that we can always change from one base to another:
$$3^x= 2^{log_2(3^x)}$$
$$= 2^{x log_2(3)}$$
and log₂(3) is just a number.

For example, I can solve the equation 3^x= 81 by taking the logarithm, base 3, of both sides: log₃(3^x)= x= log₃(81)= 4 so x= 4. (I know log₃(81)= 4 because I know 81= 3⁴ so log₃(3⁴)= 4.)

That's what log is: it's the "opposite" of the "exponential" or power function, b^x.
Of course, here I could just have written 3^x= 81= 3⁴ so, by comparing exponents, x= 4. But what if the problem were 3^x= 87 where 87 is not a simple integer power of 3? In that case, I can use either "common logarithm" (base 10) or "natural logarithm" (base e, about 2.718) to get
$$log_10(3^x)= log_10(87)$$
so
$$x log_10(3)= log_10(87)$$
and
$$x= \frac{log_10(87)}{log_10(3)}$$
The point of using "common logarithm" (or "natural logarithm") is that my calculator (in fact the calculator included with Windows that I'm running now) has both common log and natural log keys: log₁₀(87)= 1.9395192526186185246278746662244 according to my calculator, and log₁₀(3)= 0.47712125471966243729502790325512 so x is the quotient of those: x= 4.0650447521106626505692254480397, approximately, just a little larger than 4 which is what we expect since 87 is just a little larger than 81.

 

[mathwonk]

a log function is a non trivial function that changes multiplication into addition. More precisely, if f is a continuous real valued function defined for all positive reals, and if f(xy) = f(x) + f(y) for all x,y > 0, and if f(1) = 0, but f is not always zero, then f is a log functiion, and the base is the unique a>0 such that f(a) = 1.

i think this is right.

 

[cracker]

oooooohhhhh! Yeah yeah! Now I get it! LoL

 

[Data]

One of the most important "real life" applications of logarithms has to do with linearizing things. If you ever get into physics (or chemistry, for that matter), you'll run into a lot of exponentials of linear functions of various quantities. Linearizing by taking a logarithm often allows for easier curvefitting of data.

If you ever get into particle physics, you'll notice that almost everything is plotted on a log-log scale. That's done in other places too, to display large scale changes more effectively. For example, often stock performance charts will be plotted with log scales.

Not to mention thermodynamics, where one of the most fundamental quantities, entropy, is actually the logarithm of multiplicity (with an appropriate scaling factor depending on your units).

But it's always good to understand the mathematics behind things like this, before worrying about how you can use them!