Properties of Logarithms - Video Tutorials & Practice Problems

Product, Quotient, and Power Rules of Logs

Hey, everyone, when working through problems dealing with logs, you're going to come across questions that ask you to expand log expressions like say log base two of three X into multiple different logs. And we can do that using certain properties of logs. Now, don't worry, we're not just going to have to memorize a bunch of brand new rules here because all of these properties of logs actually correspond to properties of exponents that we already know and have used before. So the same way we were able to graph log functions using what we knew about exponent functions, we can do the same thing here. So I'm gonna give you a quick refresher of these exponential rules and then show you their corresponding log rule and how to use that to expand log expressions. So let's go ahead and jump right in. Now, remember when working with rules like this, we don't really care about the name. That name is not important. It's just important that we know how to use these rules and the name is just a way to organize them. So starting first with our product rule here, remember that whenever we had exponents of the same base being multiplied together, we could simply take those exponents and add them together. And we see something similar when working with logs. So if I have some log of some base with two things being multiplied together, I can separate that out into two different logs that are simply being added together. So with our exponent, we saw multiplication turn to addition and we see the same thing happening here whenever we have a terms being a multiplied in a log, this simply means that we can add two different logs together. Now we see something similar and when working with the quotient rule, because with our exponents, we saw that whenever we had exponents of the same base being divided, we would simply take those exponents and subtract them. So when seeing a log with two things being divided, what do you think I'm going to do to two separate logs? Well, I'm going to end up subtracting them. So the same way that we had division turned to subtraction with our exponent, the same thing is going to happen with our logs. So whenever we divide terms in a log, we see that we subtract two logs. So multiplication becomes addition, division becomes subtraction. So if I see something like log base two of three X like I have right here, I see this three in this X are being multiplied together. So since those things are being multiplied, I can turn this into the addition of two logs. So this would become log based two of three plus log base two of X. So see you notice that that base stays the same, but I'm simply adding two logs together and I have separated that three and that X into that addition. Now, if I'm given something like log base five of five, divided by Y since that five and that Y are being divided, this division turns to the subtraction of two separate logs. So this becomes log base five of five minus log base of why using that quota rule. Now we have one final log property to look at here the power rule. And whenever we worked with exponents, we saw that if we took an exponent of some base, so B to the power of M and raise that to another power like N I would simply end up multiplying those two powers together. Now we see something slightly different when working with our log, but it's still going to see multiplication happening. So if I have base B of M to the power of N, it's still going to turn out into multiplication. But I'm going to take this N and stick it on the front of that log. So log base M to the power of N becomes N times log base B of M. So I simply take that N and pull it out to the front and multiply it together. So anytime I'm raising a term to a power, I am simply going to end up supplying the log by the power. So we still see raising something to a power turning to multiplication. So if I'm given something like the natural log of seven to the power of two, I'm going to take that two and stick it on the front of that. And this is going to become two times the natural log of seven. So now that we've seen these log properties, we have the tools we need to expand log expressions. Thanks for watching and I'll see you in the next one.

Expand & Condense Log Expressions

Hey, everyone, we just learned all of the rules needed in order to expand log expressions. But two things are going to happen, we're going to be asked to expand much more complicated log expressions like say log base two of three XY squared. And we're also going to be asked to do the exact opposite of expanding log expressions and actually condense expressions with multiple logs down to a single log. Now, you don't have to worry because we're going to take these rules that we already know. And I'm going to walk you through how to use multiple of these rules and how to use them in their reverse direction because each of these properties can be applied in both directions depending on what your goal is, whether taking a single log to multiple logs or condensing multiple logs back down into a single log. Now you'll be able to expand and condense any log expression that gets throw your way. So let's go ahead and get started. Now, we're gonna just jump right into an example here. And in our first example, we see that we have log based two of three XY squared and we want to go ahead and expand this log expression as much as possible. So the first thing I noticed here is that I have three different things being multiplied together. This is actually three times X times Y squared. Now, since this is multiplication, that clues me in that I need to go ahead and use the product rule because I know that the product rule takes multiplication and turns it into the addition of multiple logs. So let's go ahead and expand this log out into multiple using that product rule. So this is a log base two of three plus log base two of X plus log base two of Y squared. Now I have these multiple logs, but I wanna go ahead and walk through each of these single logs to make sure that I can't expand them any further. So I have a log base two of three, I can't expand that anymore. So that's going to remain as a log base two of three. And then log base two of I also cannot expand that anymore. So that will remain the same as well. And then finally, we come to our last term log base two of Y squared. Now looking at this log, I know that I have this exponent here. And whenever I have an exponent that tells me that I can go ahead and use the power rule because the power rule tells me that I can take that exponent and pull it to the front of my log in order to expand it. So this too, I can go ahead and pull it to the front of my log. And this last term becomes two times log base two of why? Now none of these terms can be expanded anymore. So this is my final answer. Log base two of three plus log base two of X plus two times log base two of Y. We've taken this one log and expanded it out as much as we can. Now, let's look at condensing a log expression. Now, we do have to consider a couple additional things whenever we're condensing logs. And the first is that we always want to make sure that the base has to be the same. So here we have two times the natural log of X minus the natural log of X plus two. So both of these are natural logs, they have that same base of E if this was say the natural log and log base two, I couldn't condense those logs together because they have different bases. Now, one other thing we want to consider when condensing logs is that the power rule is always going to get applied first in order to get the correct answer. So with that in mind, let's go ahead and condense this log expression. So we have two times the natural log of X minus the natural log of X plus two, knowing that I need to apply the power rule. First, I'm going to go ahead and look at these terms and see how I can apply that power rule. So I have two times the natural log of X since I have this to multiplying that I know that using the power rule, I can take that thing that's multiplying it and make it into the exponent. So this becomes the natural log of X squared pulling that two into the exponent of what I'm taking the natural log of. Now, that second term I can't apply the power rule. So it's going to remain the same for now, the natural log of X plus two. Now, how else can we condense this expression? Well, I have these terms being subtracted. So since they're being subtracted, that tells me that I should go ahead and use the quotient rule because I know that using the quotient rule, I can take subtraction and I can turn it into division of a single log expression. So this becomes the natural log of X squared by what's being subtracted X plus two. So now I have taken these multiple logs and condensed it down into a single log. So this is my final answer, the natural log of X squared divided by X plus two. Now that we have seen how to expand and condense log expressions, let's go ahead and get some more practice. Thanks for watching and let me know if you have questions

Write the log expression as a single log.

$\log_2\frac{1}{9x}+2\log_23x$

• A. $\log_2x$
• B. $\log_2\frac{1}{3x}$
• C. $\log_21$
• D. $\log_23x$

Write the log expression as a single log.

$\ln\frac{3x}{y}+2\ln2y-\ln4x$

• A. $\ln\frac{3xy}{4}$
• B. $\ln\left(12x^2\right)$
• C. $\ln\left(\frac32\right)$
• D. $\ln\left(3y\right)$

Write the single logarithm as a sum or difference of logs.

$\log_3\left(\frac{\sqrt{x}}{9y^2}\right)$

• A. $2\log_3x-2-\log_39y$
• B. $\frac12\log_3x-2-2\log_3y$
• C. $\frac12\log_3x+2\log_33y$
• D. $\frac12\log_3x-2\log_39y$

Write the single logarithm as a sum or difference of logs.

$\log_5\left(\frac{5\left(2x+3\right)^2}{x^3}\right)$

• A. $5+2\log_5\left(2x+3\right)-\log_53x$
• B. $2\log_5\left(2x+3\right)-3\log_5x$
• C. $1+2\log_5\left(2x+3\right)-3\log_5x$
• D. $\log_5\left(2x+3\right)-\log_5x$

Change of Base Property

At this point, we've evaluated logs by hand and we've used different rules and properties in order to expand and condense log expressions. But sometimes you're just going to need to quickly evaluate a log by plugging it into your calculator. But if you're given something like log based pi of nine, well, that doesn't look quite so simple to just type in our calculator and go. So here, I'm going to show you that whenever you're given a log with some base that doesn't look so easy to deal with. Like say pi you can simply change the base to be whatever you need it to be. So here, I'm going to walk you through exactly how to do that. And let's not waste any time here and just jump right in. So if I have some log log base B of M, but my base B is say pi or 27 or 500 some number that is not so easy to deal with. So I want to change it, I can go ahead and change it by taking my log and turning it into a fraction. So if my original base was B, but I wanted a base of A, I would simply take log base A of M and divide it by log base A of B. So whatever I was originally taking, the log of M is going to go on the top and then original base B goes on the bottom. That's one way to remember. This base goes on at the bottom. So let's say that I'm given some log like log base five of two. And I want to change that to have a base of 10, I could go ahead and change that into a fraction and that fraction would be log base 10 of two, whatever. I was originally taking the log of divided by log base 10 of five. Now, from here, since there actually is a button for log base 10. My common log on my calculator, I can quickly and easily type this in my account calculator and come up with an answer. Now, because log base 10, your common log has a button on your calculator. And so does the natural log log base E, you're most often going to want to change your base to be either 10 or E so that you can quickly type it in your calculator and get an answer. So we just saw that whenever we changed our base to 10, this would become log base 10 of M divided by log base 10 of B. Now log base 10 can of course, just be written as long because it's the common log. So this is really just log of M divided by the log of B. Now, we see the same thing whenever we're faced with changing our base to E, we know that log base E is just the natural log. So this could simply be written as the natural log of M divided by the natural log of B. Now, you can change any log of any base to either of these two options. Most often it will be specified to you which one you want to change it into. But if it's not, you can use either one and it doesn't really matter. So with that in mind, let's go ahead and work through some examples here. So with our first example, we see log base seven of 31. So log base seven of 31 and four A and B here, we want to go ahead and use common log. So changing our base to 10. So looking at my first base here, lo log base seven of 31 I want to change this to log base 10. So doing that, I can take log base 10 in my fraction and just plug in whatever I need to plug in on the top and bottom. So I'm gonna take whatever I was originally taking the log of on top. So this is log base 10 of 31 and then divided by log base 10 of my original base seven. Now of course, log base 10 can just be written as log. So this is just log of 31 divided by the log of seven. Now, I can go ahead and type this into my calculator. And when I do, I'm going to go ahead and get an answer of 1.76. And I have fully evaluated that log. Now, let's look at another example here, we have log based pi of nine and we want to again change this into a log based 10 because we're still using common logs here. So log based pi of nine, I know that I can turn this into a log of nine divided by log of pi. So whatever I was originally taking the log of goes on the top and then of course, my base goes on the bottom. So I can go ahead and plug this into my calculator log of nine over the log of pi and I'll end up getting an answer of 1.92. And I've evaluated that log. Now looking at our third example example, C we again have log based pi of nine, but now we want to go ahead and use a natural log. So changing our base instead to E. So knowing that base E log base E is just the natural log, I know that this will turn into the natural log of nine divided by the natural log of pi. So again, whatever we were originally taking the log of goes on the top and then my original base goes on the bottom so I can go ahead and type this in my calculator natural log of nine divided by the natural log of pi and I'm going to end up getting 1.92. Now earlier, I said it doesn't matter what you change your base into. And that's because we're going to get the same answer regardless. Whenever we changed our base to 10, we got one 0.92. And then whenever we changed our base to E, we also got 1.92. So it doesn't really matter, you're going to get the same answer regardless. Let's take a look at one final example. Here we have a log base, the square root of three of E and I want to use natural logs for this one as well. So changing this to a base of E, I'm going to get the natural log of E divided by the natural log of the square root of three. Now, we can actually do some further simplification here because the natural log we know is just log base E. So this is really log base E of E. Now whenever we have the same base as what we're taking the log of, I know that this just simplifies to one. So this really just becomes one over the natural log of the square root of three. Now I can go ahead and just plug that into my calculator. And when I do, I'm going to get an answer of 1.82. And I have fully evaluated that log now that we know how to change the base of a log to be whatever we need it to be. Let's get some more practice. Let me know if you have any questions.

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.

$\log_317$

• A. 0.39
• B. 2.58
• C. 1.23
• D. 0.48

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.

$\log_967$

• A. 1.91
• B. 0.52
• C. 0.95
• D. 1.83

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log.

$\log_841$

• A. 1.61
• B. 0.9
• C. 0.56
• D. 1.79

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log. 

$\log_23789$

• A. 0.08
• B. 11.89
• C. 3.58
• D. 0.30