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## Algebra 2 (Eureka Math/EngageNY)

### Unit 3: Lesson 4

Topic B: Lessons 7-9: Logarithms intro# Intro to logarithms

CCSS.Math:

Sal explains what logarithms are and gives a few examples of finding logarithms. Created by Sal Khan.

## Want to join the conversation?

- Is log base 1 of 1 equal to 1, 0, or both?(3 votes)
- Log base 1 of 1 would have infinite answers not just 1 and 0(5 votes)

- Can a logarithm be written without a base?(5 votes)
- Writing a log without a base is implied that it has base 10.

log(100) = log_10(100) = 2

The other "exception" is the natural log, which you don't write with a base.

ln(x) is the same form as log_e(x), but ln is more perferred.(7 votes)

- In what grade do you learn logarithims?

Thanks in advance.(2 votes)- Hi! I learned it in Algebra 2, which normally people take in 10th grade. I took it in 9th grade.(3 votes)

- what is the value of n if 0.28=1/(1.09)^n? using logarithm(2 votes)
- n=log_(1/1.09)=0.28 is the logarithmic form(2 votes)

- is there any synonym of logarithms?(4 votes)
- Yes! And explain how to use logarithms with decimals!(4 votes)
- Is the word 'base' used in a logarithm context with a different meaning than when one says that a number is written in base ten, base two, base five, etc.?(2 votes)
- The word base refers to its definition as used with exponents.

If you have 2^6

The "2" is the base and the "6" is the exponent.

"Log base 2 (64)" is asking you to find what exponent would you apply to the base "2" to create 64. Or... 2^( _ ) = 64. fill in the appropriate exponent.

Hope this helps.(4 votes)

- can you teach my divison?(2 votes)
- Start learning it with a times table. To multiply on a times table (such as 6*8) find there the six line meets the 8 line at 48. To divide 48/6, go on the 6 line until you find 48, then go up to 8. To divide 48/8, go on the 8 line until you find 48, then go up to 6. This should help get the idea of multiplication and division being opposite operations.(4 votes)

- Because a^2, for example, is axa, but can also be defined as 1xaxa. Continuing down with this, you get a^1 = 1xa and then a^0 = 1. Also, there is a pattern that going down an exponent for a would be dividing by a. So a^1 = a, and then a/a = 1. For more explanation, try this link: https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-arithmetic-operations/cc-6th-exponents/v/the-zeroth-power(1 vote)

- Who invented logarithms? And for what reason?(2 votes)
- "logarithm" is his word for "exponent." I really helps when you are looking at exponential equations such as a^x=b where there is a placeholder in the exponent. We can read this equation so: "x is the exponent (logarithm) to the base 'a' that will give us 'b.'" We can write this in logarithmic for like so: x=log_a(b). This is much easier to solve than a^x=b, especially since we can do very complicate problems very simply.

NOTE: when you just see plain old "log(b)," the _a value is 10; this is know as "the base ten logarithmic scale" for which there are charts of logarithms for numbers 1.0000 to 9.9999.(3 votes)

## Video transcript

Let's learn a little bit about the wonderful world of logarithms. So we already know how to take exponents. If I were to say 2 to the fourth power, what does that mean? Well that means 2 times 2 times 2 times 2. 2 multiplied or repeatedly multiplied 4 times, and so this is going to be 2 times 2 is 4 times 2 is 8, times 2 is 16. But what if we think about things in another way. We know that we get to 16 when we raise 2 to some power but we want to know what that power is. So for example, let's say that I start with 2, and I say I'm raising it to some power, what does that power have to be to get 16? Well we just figured that out. 'X' would have to be 4. And this is what logarithms are fundamentally about, figuring out what power you have to raise to, to get another number. Now the way that we would denote this with logarithm notation is we would say, log, base-- actually let me make it a little bit more colourful. Log, base 2-- I'll do this 2 in blue... Log, base 2, of 16 is equal to what, or is equal in this case since we have the 'x' there, is equal to 'x'? This and this are completely equivalent statements. This is saying "hey well if I take 2 to some 'x' power I get 16'." This is saying, "what power do I need to raise 2 to, to get 16 and I'm going to set that to be equal to 'x'." And you'll say, "well you have to raise it to the fourth power and once again 'x' is equal to 4. So with that out of the way let's try more examples of evaluating logarithmic expressions. Let's say you had... log, base 3, of 81. What would this evaluate to? Well this is a reminder, this evaluates to the power we have to raise 3 to, to get to 81. So if you want to, you can set this to be equal to an 'x', and you can restate this equation as, 3 to the 'x' power, is equal to 81. Why is a logarithm useful? And you'll see that it has very interesting properties later on. But you didn't necessarily have to use algebra. To do it this way, to say that 'x' is the power you raise 3 to to get to 81, you had to use algebra here, while with just a straight up logarithmic expression, you didn't really have to use any algebra, we didn't have to say that it was equal to 'x', we could just say that this evaluates to the power I need to raise 3 to to get to 81. The power I need to raise 3 to to get to 81. Well what power do you have to raise 3 to to get to 81? Well let's experiment a little bit, so 3 to the first power is just 3, 3 to the second power is 9, 3 to the third power is 27, 3 to the fourth power, 27 times 3 is equal to 81. 3 to the fourth power is equal to 81. 'X' is equal to four. So we could say... Log, base 3, of 81, is equal to-- I'll do this in a different colour. Is equal to 4. Let's do several more of these examples and I really encourage you to give a shot on your own and hopefully you'll get the hang of it. So let's try a larger number, let's say we want to take log, base 6, of 216. What will this evaluate to? Well we're asking ourselves, "what power do we have to raise 6 to, to get 216?" 6 to the first power is 6, 6 to the second power is 36, 36 times 6 is 216. This is equal to 216. So this is 6 to the third power is equal to 216. So if someone says "what power do I have to raise 6 to-- this base here, to get to 216?" Well that's just going to be equal to 3. 6 to the third power is equal to 216. Let's try another one. Let's say I had, I dunno, log, base 2, of 64. So what does this evaluate to? Well once again we're asking ourselves, "well this will evaluate to the exponent that I have to raise this base 2, and you do this as a little subscript right here. The exponent that I have to raise 2 to, to get to 64." So 2 to the first power is 2, 2 to the second power is 4, 8, 16, 32, 64. So this right over here is 2 to the sixth power, is equal to 64. So when you evaluate this expression you say "what power do I have to raise 2 to, to get to 64?" Well I have to raise to the sixth power. Let's do a slightly more straightforward one, or maybe this will be less straightforward depending on how you view it. What is log, base 100, of 1? I'll let you think about that for a second. 100 is a subscript so it's, log, base 100, of 1. That's one way to think about it, I'll put parentheses around the 1. What does this evaluate to? Well this is asking ourselves, or we would evaluate this as, "what power do I have to raise 100 to, to get to 1?" So let me write this down as an equation. If I set this to be equal to 'x', this is literally saying 100, to what power, is equal to 1? Well anything that a 0 power is equal to 1. So in this case 'x' is equal to 0. So log, base 100, of 1, is going to be equal to 0. Log base anything of 1, is going to be equal to 0 because anything to the 0 power and we're not talking about 0 here. Anything that is to the power of 0 that is not 0, is going to be equal to 1.