Logarithms and what they are for.

[jasonlr82794]

The math behind the logarithms isn't hard especially if you can remember where things go and how the math is worked out but what are they for? What do they do and why are they used? Maybe I am missing what they are used for in the math but it really just doesn't make sense. You have a base number and have to figure out what the exponential number is but why use this?

 

[Mark44]

The natural log function, ln, is the inverse of the natural exponential function. If y = e^x, then x = ln(y). Both equations describe exactly the same pairs of numbers, meaning that any point (x, y) on the graph of either is also on the other graph.

A practical reason for using logarithms is to narrow a range of values that differs by many orders of magnitude. A function f that grows exponentially exhibits this behavior. Instead of graphing the exponential function y = f(x), it can be helpful to graph the log of the function values; i.e., y₁ = ln(f(x)), which will yield a straight-line graph. This is the reason behind semi-log graph paper, which can be used to determine a formula for data that doesn't fit a line or low-degree polynomial.

 

[jasonlr82794]

So in the first paragraph you are saying that those two functions graphed out are the inverses which will look like a regular inverse and noninverse function on the graph? The second paragraph is kind of vague to me, could you add some math in that?

 

[Mark44]

So in the first paragraph you are saying that those two functions graphed out are the inverses which will look like a regular inverse and noninverse function on the graph?

No, I am saying that there is only one graph. You can call it y = e^x or you can call it x = ln(y). If a pair of numbers (x, y) is on the first graph, exactly the same pair is on the other graph.

I don't know what you mean by "regular inverse" and "noninverse" functions.

The second paragraph is kind of vague to me, could you add some math in that?

For an example of what I was talking about, consider this set of data:
{(1, .03), (2, .3), (3, 3), (4, 30), (5, 300), (6, 3000)}

You would be hard-pressed to graph this data (which I cooked up as an example). If you graphed this on semi-log graph paper, which has an ordinary scale along one edge and is logarithmic along the other edge, you could graph this data easily, and could determine a formula that fits it.

 

[berkeman]

The math behind the logarithms isn't hard especially if you can remember where things go and how the math is worked out but what are they for? What do they do and why are they used? Maybe I am missing what they are used for in the math but it really just doesn't make sense. You have a base number and have to figure out what the exponential number is but why use this?

As Mark44 said, they are often useful when dealing with quantities that vary over a very wide range of values. For example, in electronics, we often deal with filters and transfer functions of amplifiers, where the frequency range is very wide (like 1Hz to 100MHz), and the amplitude response can be very wide as well (like a few volts RMS down to 1uVrms). The most practical way to deal with those kinds of wide ranges is to take the log of the values, to give you more reasonable numbers to deal with.

In engineering, we use the "decibel" (dB) definition to use logarithms to deal with these wide ranges. For voltages, we make this definition:

V[dBV] = 20 log (V/1V)

or

V[dBuV] = 20 log (V/1uV)

So 1Vpp is the same as 0dBV, or +60dBuV. 1uVpp is the same as 0dBuV or -60dBV.

Using logarithms let's us deal with a factor of 60, instead of a factor of a million in the above examples.

 

[SteamKing]

Logarithms were used for many years before the invention of mechanical and electronic calculators to simplify arithmetic calculations. For example, knowing the logarithm of two numbers, one could multiply them together by adding their logarithms, and then using this sum to find the product by using a table of logarithm values. Similarly, taking the logarithm of a number and multiplying it by 2 could be used to find the square of a number; dividing the logarithm by 2 could be used to find the square root. Slide rules used a logarithmic scale in order to find products, quotients, powers, roots, etc.

In many textbooks printed before 1970 or so, one of the standard appendices included in the back was a table of logarithms to help students with the calculations needed to solve the problems in the book. All of these tables have since disappeared (like slide rules) because inexpensive calculators are so common.

 

[Borek]

Ever heard about thing called pH - which describes acidity of the solutions? Well, it is -log(H⁺ concentration). pH of a solution of a strong acid is typically around 0 or 1 (which means concentration of H⁺ is 1 or 0.1 mol/L), pH in your stomach is somewhere bettween 2 and 3 (0.01-0.001 mol/L), beer is around 3-4 (10^-3-10^-4 mol/L), pure water should be around 7 but is usually slightly more acidic with pH between 5-6, your blood keeps pH of 7.3, bleach has pH of 12 and so on.

pH not only uses logarithm because it was defined this way, turns out in many cases real world answers linearly to pH. If you take a glass electrode (used for pH measurements in pH meters) and you plug it into water, it will take potential that is directly proportional to pH - not to the H⁺ concentration. So apparently we have not "invented" logarithms, we just "rediscovered" them long after real world was using them .

 

[arildno]

In every physical case where two quantities are related in such a way that the increase of quantity A is proportional to the relative increase in quantity B, logarithms will be a natural way to describe the relationship.

Borek has meantioned one such case, there are numbers of others.

For example, for perceptions, say perception of load you carry, the heightening of perception of carried load will typically be proportional to the relative increase of the load you carry, rather than proportional to the absolute increase of the load you carry.

If you already carry 25 kg, you won't even perceive a load increase of 10 grams, something you certainly would do if you only carried 5 grams prior to the increase.