The logarithmic base ten form represents the exponent that 10 must be raised to in order to obtain that specific number. Now, what does this really mean? Let’s take a look at these examples here. So here we are looking at the power that we can raise 10 to and the answer that results. So 101 is just 10 times 1 which gives us 10. 104 is just 10 times 10 times 10 times 10 which gives us 10,000. 10-1 is equivalent to just saying 1 over 10. Right? Because what it’s really saying is 10 inverse, so that’s 1 over 10, so that gives us 0.10. And then 100, any number to the zeroth power, equals 1.

How does this relate to my log function? Well here we have log(ten) gives us 1. Really what this is saying, it’s saying that 10 to what number gives me 1? The answer would be 1. Here, log of 10,000. Alright, so think of it like this, we have log of 10,000 so that is really log(104). Whatever the number is here, because remember this is log base 10, what happens here is that this cancel out with this so we get 4 left at the end.

Here when we get log of 0.10. What does that really mean? That really means log(10-1). So this cancels out with this and gives us negative one. Then we have log of 1. Right? So you’d say here, so log base 10 and then 10 to the one here, well, well, 10 to the 0 here actually sorry, equals 0.

So that is what is really going on here. And if we have log(ten) think that as log base ten and then 101 equals 1. So we’re just converting each of these values, by 10 to some power. The log portion cancels out the 10 and leaves behind the exponent as my final answer. That’s how we can calculate log. I know you guys have calculators, but there may come a point where you have this within a math class or within an MCAT or PCAT or OAT or DAT later on, much later on after you’ve taken all these science courses where you have to understand these relationships. And this is how our log base function is connected to multiples of 10. Knowing that helps us to get to the answer.

Now understanding this, try to see if you can answer example 1. So here we are going to use this without a calculator. Try to do it without a calculator, see what answer you get, and then afterwards come back and see if your answer matches up. I’ll help you guys a little bit for the first one. So here, this log is getting distributed to the 1 and to the 10-7. Now when that happens, what that means is we have log of 1, and because they’re multiplying it really means that we’re adding, So it’s plus log of 10-7. See if you guys can figure out what the answer is without using a calculator. Come back and see if your answer matches up with mine.