Background Information on Exponentials and Logarithms


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Since the treatment of the decay of radioactive nuclei is inextricably linked to the mathematics of exponentials and logarithms, it is important that you have some expertise in using them. This is intended as a brief outline of how to use them. Note that you calculator will have these operations built into it, allowing you to get the result of the operation with a single key stroke.
We might note in passing that the mathematics below is not all limited to the topic in the class, that is radioactive decay. It is not even limited to physics or the other sciences. For example if you establish an account which pays compound interest at a fixed rate, and reinvest the interest into the account, then the value of the account increases exponentially with time, following all of the same mathematical formulae below.

Step 1 - Integral Powers of 10

Let us start with something with which (I hope) you are already familiar, that is powers of 10. The notation 10x means 10 multiplied by itself x times,

Step 2 - NonIntegral Powers of 10

Let us take the previous idea on step further, suppose that the power x is not a whole number, say 4.518. Then what is 10x now? You will need your calculator to solve this one. Depending on your calculator the exact keystrokes will be different, but there is a good chance that there is a key labeled 10x. Try entering 4.518 followed by 10x, and you should get the result 104.518 = 32960.97

Step 3 - Base 10 Logarithms

The previous step solved the equation y = 10x, if you are given x then you can calculate y. If x = 4.518 then y = 32960.97. However, suppose you are given y and are required to calculate x. For example what is the value of x if y = 69. As an equation solve for the variable x if 69 = 10x.
The key to solving this equation is the logarithm (abbreviated to just log). It is defined as the opposite operation to the power of 10:

If y = 10x then x = log y

Again your calculator probably has a key labeled log which performs this operation for you. Type in 69 followed by log, and you should get the result 1.838849. Just a check try using the method above and you should get the result 101.838849 = 69. (It might not be exactly 69 because the number 1.838849 has been rounded off a little, but it should be very close.)
You might want to try the following for practice:

Step 4 - Negative powers

In all the examples above x was a positive number, in which case 10x is always greater than 1. The value of x could be negative, say x = -0.55. In that case y = 10x is less than 1. Your calculator can solve these in the same manner. Type in 0.55, then the change sign key, and then the 10x key. You should get the result 0.281838.

You might want to try the following for practice:

Negative powers are handled by your calculator also. Suppose we want to solve 0.01234 = 10x. Rearranging the equation using the definition of the logarithm from above, this becomes x = log0.01234. Try typing in 0.01234 into your calculator and press the log key. You should get the answer x = -1.90868

You might want to try the following for practice:

Step 5 - Dealing with bases other than 10

In all the above expressions we have used base 10, that is y = 10x. However there is nothing special about base 10, except that it is the base that you are used to using, the decimal system. We could have used any base number such as 2 (the binary system) or 8 (octal system) or 16 hexadecimal system, or 11, or anything else Again these work even if the exponent is not an integer

Step 6 - Non integral bases

So far we have considered the case when the power is not an integer. Can we do the same if the base is not an integer? The answer is yes.

Step 7 - e

One of these non-integer numbers is so special in mathematics it is given its own symbol, e which stands for 2.7182818284590452353602874713527...... (The reasons why it is so special have to do with calculus and need not concern us here.) However, it in all respects it behaves just like all other bases

Step 8 - Putting powers and logarithms together

A simple rule is that, for the same base, raising the base to a power and finding the logarithm are opposite actions. One followed by the other has no overall effect Note though that this rule does not work if the bases are different. For example ln(10³) is not three. The power operation used base 10, but the logarithm used base e.
 

Step 9 - A radioactive half life problem

The equation which describes the radioactive decay of an nucleus is N = No e-kt where No is the initial number of nuclei, N is the number remaining after a time t, and k is a constant related to the lifetime (k = ln2/T = 0.6931/T). With a little bit of rearranging this can be written as y = ex if y stands for the fraction N/No and x stands for -kt. Using the rules developed above we can solve these radioactive decay problems. As mentioned above this mathematical treatment relates to similar problems in many walks of life. Let's go through the following examples:
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