Understanding Logs Intuitively PDF

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Summary

Are you frustrated with working with logs without even knowing what they mean? Here's a lecture I chalked to help you understand logs intuitively including what they are, how they work, and how they tie in with the rest of your knowledge about math....

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Understanding Logs Intuitively The Learning Center Georgia College & State University

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Introduction

The goal of this document is to build your intuition of what logs are, how we work with them to solve problems, and how they play their part in algebra. If you’re tired of just going through the motions of solving equations with logs, and you actually want to see why the rules make sense, keep reading. After reading this, we hope you’ll be able to go further in understanding these functions into seeing their great importance in all fields of mathematics! Before we get into it, recall first how exponents work. Take this equation: 32 = 9. Make’s sense, right? We know that the exponent of 2 makes us have to take that 3 and multiply 2 of them together to get 9. But this thought process comes to us because we’re used to reading from left to right. Think about it, we started with the 3, saw the 2, did what we needed to do to that 3 because of the 2, and came to the conclusion that the 9 on the other side made sense. But what if we rearranged this equation using logs, so that the numbers appear in a different order? Let’s look at it: log3 (9) = 2. This is another way of expressing the previous equation using logs. The two expressions are equivalent in meaning, but the logic behind them is completely different. It’s almost like we’re reading the same sentence in two different languages, of which their sentence structures are different, and this is important to understand. Let’s try reading the logic from left to right this time. 1. ”log base 3” : Ok, we’re going to be talking about of 3. 2. ”9” : This is the input to the log of base 3. 3. ”=” : Alright, on the other side of this equal sign must be the exponent that we put on 3 to get 9. The other side of the equal sign must be how many times we multiply 3 by itself to arrive at 9. 4. ”2” : There it is. Multiplying 3 2 times will give us 9. So the statement is true!

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Now, we’ve worked our way through this representation. The logic is way more jumbled this way, but the importance lies with the destination. Before, we asked the question, ”What does multiplying the 3 by itself 2 times give us?” Now, we ask the question, ”What exponent should we put on 3 to give us 9?” If you’ve been paying attention, you might notice that the order of the numbers in each of these expressions changed. In the first we started with 3, saw the 2, saw what an input of 2 as the exponent on 3 did, and arrived at 9. In the second, we started with the base 3, saw the input of 9 in the log, saw what the log of base 3 did to that 9, and arrived at two. The inputs and outputs of the two expressions seemed to switch. This is because exponential functions and logarithmic functions are inverse functions!

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The Question You Should Be Asking

From our knowledge of functions, we know that every valid input into a function gives exactly one output. So when given a specific function, we can ask ourselves a unique question on what we should do with our input to help us arrive at our output. With exponential functions, we ask, ”what does multiplying our base with itself [input] amount of times give us?” With logarithms, we ask, ”What exponent do I put on the base number to arrive at my [input]?” This question is a little backwards in our minds at first, but it’s because we’ve gotten used to working with exponents (hopefully) before we got to logarithms. But the use of this question, the introduction of logarithms does a lot more than give us more problems to solve. It rearranges the inputs and outputs so that we are now looking at exponents more closely. The goal of this document is this: to look at logs as making us shift our minds to not looking at how bases numbers work, but looking at how exponents work. If you haven’t noticed yet, the log rules are the EXACT SAME as the rules of exponents! Exactly the same, except that we’re looking through a different lens. Hopefully looking at exponent rules and log rules together will illustrate this...

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Understanding the Rules

What happens when you multiply two terms with exponents together if they have the same base? 23 ∗ 24 =? If we look at them as exponential functions, we can take the exponents on the 2’s as inputs. What do we do with those inputs? Well, we add them: 23 ∗ 24 = 27 . 2

Visually, it’s kind of like the acted of multiplying the bottom stuff, the base numbers together translated into doing a different operation, a smaller operation to the numbers on the top, the exponents. Multiplying the whole things together, means the exact same thing as simply adding the numbers on the top. I don’t mean to kid you on this, it’s just this is where the intuition lies. The truth lies here: 23 =8 24 =16 27 =128 23 ∗ 24 =27 (8) ∗ (16) =(128) 128 =128. Now let’s look at the log rule: logx M + logx N = logx M ∗ N. What happened? Why does adding two logs with the same base make us multiply the inputs together? Well, you can just as easily ask why did we take two exponential functions of the same base and add their inputs, the exponents together? Because it’s the exact same thing! The only difference is their representations. Consider the log rule. Let x = 2, and say M = 8 and N = 16. You might have already guessed what’s going to happen, but let’s look: log2 8 + log2 16 = log2 8 ∗ 16. What now? Well, ask the question, ”What exponent do I put on the base number 2, to arrive at my inputs, 8 and 16?” With a little bit of computation, we see that they are 3 and 4, respectively. On the other side, 8 ∗ 16 = 128, and the exponent on 2 that gives us 128 is 7. So we have log2 8 + log2 16 = log2 8 ∗ 16 3 + 4 =7. This is the intuition: that each log term IS the exponent. The term log2 (8) IS the exponent 3. The term log2 (16) IS the exponent 4. When we use logs, we’re talking about exponents, and we’re shifting our focus from the world of bases, from normal numbers that we’ve been working with since we started doing math to the realm of exponents. The same can be seen with the division rule of exponents: 24 ÷ 23 = 24−3 = 21 . With this rule of exponents, when we divide the whole term 24 by 23 , we take the 2, subtract 3 from 4, and are left with 21 . The same goes with the the outputs of the logs: log2 16 − log2 8 = log2 2 3

. The above statement is true, and we can see that once we see what each individual term evaluates to. log2 16 =4 log2 8 =3 log2 2 =1 log2 16 − log2 8 = log2 2 4 − 3 =1. It is a misstep to not think of log terms as exponents. When we see a log, we should shift our focus from the whole term, the 23 or the 24 to the exponents themselves 3 and 4. This is the essence of logarithmic functions. They should shift our focus from base numbers to exponents, and so when you see a log term, you should see an exponent.

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Steps on a Ladder

Looking at logs as exponents opens up some interesting doors, and brings light to how exponents and base numbers relate to each other. What I mean is this: they fall into a certain hierarchy, their position in which determines their power on the value of a number. Remember how multiplication is repeated addition? The first product tells you how many times you add the second number together, or however you want to do it. And further how exponentiation is repeated multiplication? We know by now these operations that as we go from adding to multiplying to putting a power on something is increasing the power of our influence on a number, we can make the value of a number shoot off by going up in these operations. There’s a natural hierarchy of power with them. You can think of it as a step ladder with each step being one of the operations, ordered by their power of influence on numbers. So the first step is addition, the second multiplication, and the third is exponentiation. Imagine that if you were to throw two numbers on the same stair, you’d have to use the operation of that step to combine the two. Like if you through a 2 and a 4 on the multiply step then you’d multiply them to get 8. Having this setup allows us to see an analogous hierarchy about the types of numbers that we use operations to combine. Base numbers and logs have this same division in their power of influence on values, but where the power of operations is shown through how they combine numbers, the power of base numbers and logs is shown through how much the end product grows after two of the same kind combine. Think about it. If you take two base numbers and throw them on the first step, you’d add them together. The two numbers, these are the inputs, combined to be a sum, the output. This doesn’t really show what we’re talking about, but let’s think about what happens when you do this with logs. Throwing two logs on the first step makes you add them also, but what does this mean? If we have a log, we have to have in our minds two numbers, the base and the exponent, since the exponent isn’t an exponent if it doesn’t have a base. It’s kind of sweet. But the point is, when two logs, let’s consider log2 2 and log2 8, are combined through 4

addition, the log rule for addition means that the values of the inputs to the logs must multiply together resulting in log2 16. It’s as if the inputs to the logs, in this sense the 2 and the 8 (which if you think about it is the output of the the numbers in exponential form, like 24 = 16), are on the second step of the ladder. What happens when we throw two logs on the second step? The log rule of multiplication tells us that you can take a number, say 3, and when you multiply to a log, the value of the end result will be equal to what you get if instead you decided to put that 3 as the exponent of the log input. Well that’s like saying those two numbers, the original input and the 3 were on the third step. (I realize this is a little bit different with the previous situation since the 3 doesn’t itself change but it’s role in the scenario changes. But you can think about focusing on the input of one log, and seeing how the other number whether it be a log or just a base number influences its value like it’s acting on it, being its exponent gives an appropriate shift in perspective.) So what happened? when the outputs of the logs were on the first step, the inputs acted like they were on the second step. And when the former was on the second, the latter was on the third. Looking at this relationship of the input and outputs of logs, we can see that the outputs, which are the true value of the logs are always one step lower than the outputs. If two outputs were on one step, the inputs would be on the one above. Do you see the hierarchy? There’s a tendency to say that the outputs, the actual value of the logs, is lower on the hierarchy since they are one step lower on the ladder, but this is very misleading. The existence of hierarchies is built on the relationships between the groups of the hierarchy; we shouldn’t be thinking about where the inputs and outputs are on the step ladder, but how the location of one group on the ladder influences the location, the power of influence of values on the others. It is a misstep to not think of exponents when we think of logs, and this step ladder idea illustrates the importance of that. Logs, exponents, have a higher power of influence on base numbers, on their inputs. And all the log rules in algebra reflect that.

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