Logarithms

A contagious virus enters a society. Disease starts to spread. Every day, twice as many people are infected as the day before, that is, two times as many. Today 1 person is infected. Tomorrow 2 people are infected.

After two days, 2 times 2 people are infected. This is 2 to the power of 2. After three days, 2 times 2 times 2 people are infected. 2 to the power of 3. And so on.

There are twice as many new infections every day. How many days will it take until 32 people are infected? 2 times 2 Times 2 Times … How many twos do we have to multiply by before the result is 32? In other words, what should 2 be raised to the power of, to give the result 32? 2 to the power of something, which we call x, equals 32.

What is this something, or this x? First we should remind ourselves that 2 is the base in the number 2 to the power of x. 32 is the result. The number we are looking for, the number x, we call the base-2-logarithm of 32. The base-2-logarithm of 32 equals x.

Now, let’s return to the virus and the number of infections. After three days, 2 times 2 times 2 people are infected. 2 to the power of 3. This is 8. After four days, 2 to the power of 4 people are infected.

This is 16. After five days, 2 to the power of 5 people are infected. This is … 32! There it is! Now we know that the base-2-logarithm of 32 equals 5.

This is a logarithm. And the number, the base-2-logarithm of 32, is written in logarithmic form. Here is another number. A base, which is 2 again, and an exponent, 3. This number is written in exponential form.

The exponential form and the logarithmic form are related. Watch this: 2 to the power of 3, that is 2 times 2 times 2, equals 8. The base-2-logarithm of 8 equals 3. Because 3 is the number 2 should be raised to the power of to get the result 8. Look at the expressions.

Can you see that the numbers 2, 3 and 8 are there, whatever form you choose? 2 is the base. 3 is the exponent in the exponential form, and the result of the logarithm. 8 is the result of the exponential expression, and the number inside the logarithm. This is the relation between the exponential form and the logarithmic form.

The logarithm answers the question: What should a certain base be raised to the power of, to get a certain result? Let’s look at another example. 3 to the power of x equals 81. Now we are searching for the value of the exponent x. And we get the answer from … a logarithm.

The base in the exponential expression is 3. So we look for a base-3-logarithm. And the result of this exponential expression is 81. So we are searching for the base-3-logarithm of 81. Let's try some calculations.

3 times 3 is 9. 3 times 3 times 3 is 27. 3 times 3 times 3 times 3 is … 81. 3 times itself 4 times! 3 to the power of 4.

So, the base-3-logarithm of 81 equals [sneezes] - 4! Oh great! Now I’m infected too!