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### Course: Algebra (all content)>Unit 11

Lesson 22: Introduction to logarithms (Algebra 2 level)

# Relationship between exponentials & logarithms: tables

Given incomplete tables of values of b^x and its corresponding inverse function, log_b(y), Sal uses the inverse relationship of the functions to fill in the missing values. Created by Sal Khan.

## Want to join the conversation?

• Is it necessary to include parenthesis around the base number?
• It is required sometimes, but you can write it whatever way you want on your own.
• In the table, b is proved to be 2. We all know that 2 raised to power 3 is 8. In the table, 2 raised to power 2.822 is 7, close to 8. But the next power, 2.169, is less than 2.822 but still produced the number 9. How is it possible? Is there some kind of mistake or what? Was it 3, accidentally written as 2?
• I think so.
(1 vote)
• How would you calculate a decimal exponent?
• You will first have to break up the exponents so that you will reduce the number of numbers with decimals as exponents. Solve the numbers with integral exponents. Then, turn the decimals exponents to fraction exponents. Let's say the number is 4^(1/2). Then you would take the square root (since it is 1/2) of the base. If it was 4^(1/3), then you would take the cube root and so on. Please let me know if I have answered your question.
• How would you graph these?
• There is a video called "Plotting points of logarithmic function". I'd check it out first and see if that clears anything up.(:
• I think the number in the first graph's fifth column is wrong.
2^2.169 is 4.497, 2^3.169 is 8.994, so it should be 3.169 instead of 2.169.
• When solving for c, I got to 2^1.585=2c, and then divided both sides by 2, so I had: (2^1.585)/(2^1) = c. Then I just got 2^0.585 = c by subtracting the exponents. I know that we don't know what 2^0.585 equals without a calculator, but is that a valid approach?
• Yes this is a valid approach. If you wanted to quickly test your answer you could have found 2^0.585 in your calculator and then compared it to (2^1.585)/(2/1)
• Not sure how in the first table 2 raise to 2.169 equals to 9 (Last column) .. It should be 4.5 isn't it? Or am I missing something
• You missed the pop-up box in the lower right-hand corner that appears at which indicates that the entry should be 3.169, not 2.169.
(Note that 2 TIMES 4.5 is 9, but in this problem we are raising 2 to the POWER of 3.169.)
• - why are both sides divided by 10 rather than 5?
(1 vote)
• basic alegbra.
• On the left column the 2.169 should be 3.169 right?
• I agree with you. Why not report this to the Khan Academy as a "mistake" in the problem?
• You're really clear!I paused it and solved it.

## Video transcript

Voiceover:We've got 2 tables over here. This first table tells us that for any given value of X, what is the value of b to the x? For example, if we look right over here, if x is 1.585, b to the 1.585 is 3. This is telling us that b to the 1.585 is equal to 3. Similarly ... I can never say that ... it is telling us that b to the 2.322 is 5. b to the 2.807 is 7. b to the 2.169 is 9. Now this table over here is telling us for any given value of y, what is going to be log base b of y? This tells us that log base b of a is 0. Log base b of 2 is 1. Log base b of 2c is 1.585. Log base b of 10d, so this is literally, this is telling us that log base b of 10d is equal to 2.322. That's what this last column tells us. Now what I challenge you to do is pause this video, and using just the information here, and you don't need a calculator; in fact, you can't use a calculator. I forbid you; try to figure out what a, b, c, and d are just using your powers of reasoning, no calculator involved. Just use your powers of reasoning. Can you figure out what a, b, c, and d are? I'm assuming you've given a go at it, so let's see what we can deduce from this. Here, we have just a bunch of numbers. We need to figure out what b is. These are all b to the 1.585 power is 3. I don't really know what to make sense of this stuff here. Maybe this table will help us. This first ... Let me do these in different colors. This first column right over here tells us that log base b of a, so now y is equal to a, that that is equal to 0. Now this is an equivalent statement to saying that b to the a power is equal to ... oh sorry, not b to the a power. This is an equivalent statement to saying b to the 0 power is equal to a. This is saying what exponent do I need to raise b to to get a? You raise it to the 0 power. This is saying b to the 0 power is equal to a. Now what is anything to the 0 power, assuming that it's not 0? If we're assuming that b is not 0, if we're assuming that b is not 0, so we're going to assume that, and we can assume, and I think that's a safe assumption because where we're raising b to all of these other powers, we're getting a non-0 value. Since we know that b is not 0, anything with a 0 power is going to be 1. This tells us that a is equal to 1. We got one figured out. a is equal to 1. Now let's look at this next piece of information right over here. What does that tell us? That tells us that log base b of 2 is equal to 1. This is equivalent to saying the power that I needed to raise b to get to to 2 is 1. Or if I want to write in exponential form, I could write this as saying that b to the first power is equal to 2. I'm raising something to the first power and I'm getting 2? What is this thing? That means that b must be 2. 2 to the first power is 2. So b is equal to 2. b to the first power is equal to 2. You could say b to the first is equal to 2 to the first. That's also equal to 2. So b must be equal to 2. We've been able to figure that out. This is a 2 right over here. It actually makes sense. 2 to the 1.585 power, yeah, that feels right, that that's about 3. Now let's see what else we can do. Let's see if we can figure out c. Let's look at this column. Let's see what this column is telling us. That column we could read as log base b. Now our y is 2c. Log base b of 2c is equal to 1.585. Or we could read this as b, if we write in exponential form, b to the 1.585 is equal to 2c. Now what's b to the 1.585? They tell us right over here that b to the 1.585 is 3, is 3, so this right over here is equal to 3. We get 2c is equal to 3, or divide both sides by 2, we would get c is equal to 1.5. This is working out pretty well. Now we have this last column, which I will circle in purple, and we can write this as log base b of 10d is equal to 2.322. This is saying the power I need to raise b to to get to 10d is 2.322, or in exponential form, b to the 2.322 is equal to 10d. Now what is b to the 2.322? They tell us over here. b to the 2.322 is 5. This is equal to 5. We could write 10d is equal to 5, or divide both sides by 10. d is equal to 0.5, and we're done. We were able to figure out what a, b, c, and d are without the use of a calculator.