Question 1

Hi,

This might be a silly question but how is y = log_b (X) the inverse of y=b^x? Isn't the inverse of y=b^x this log_b (y) = X?

Thanks

Accepted Answer

I guess if you look at the function as y = f(x) = b^x. The inverse of the function would be x = f(y) = b^y. So, the logarithmic form of the inverse function is  log_b(x) = y.  
Ey that's just my take, might be wrong too lol.

Question 2

How did he know to use points 1,4, and 16 in the second chart? I thought that, given the question, he would have taken the x values from the existing points and plugged them into the 2nd function.

Accepted Answer

For anyone who watches this video and has a similar question, think about it this way: 
y = b^x can be rewritten as log_b(y) = x.

So for b = 4 (you can deduce that from the graph) and y = 1, 4, 16 (also deduced from the graph) you have

log_4(1) = 0
log_4(4) = 1
log_4(16) = 2

Which represent points (1, 0), (4, 1) and (16,2) the inverse of the points on the given graph!

Question 3

at around 1:40, why is sal taking the y values of the first table and using them as the x values of the second table?

Accepted Answer

I had a little trouble on this too, but here's what I figured: The *purpose of an exponential function is to  find what value a number equals when it is raised to the power of x*.  However, the *purpose of a logarithm is to find the exponent (x value)* that gives you a certain amount when you raise a base by the unknown x value. Though it might sound complicated, if you try to define the purpose yourself, you'll end up with the same answer. For this reason,  they are like the inverse functions of each other, just like multiplication and division.

In other words, the logarithm tries to lead you to the exponent needed to reach the value, while the exponential graph tries to lead you to the value given by the exponent's use.

Therefore, they are inverse operations as they undo each other. Furthermore, they reflect over the y=x line, and their coordinates are switched.  If you check at 3:29, you'll see Sal agrees. If you check the background at this time, you'll also see how when the b= 4, everything fits together perfectly.

Hope this helps.

Question 4

At 0:44, how do we know that b raised to the power 1 is equal to four. Is it a definite variable with its value

Accepted Answer

It is a constant that we deduced from the graph that is given to us.

Question 5

Sorry about this but I have two questions. The first might be silly, but here goes: How does one solve for x when it is an exponent such as 2^x=1/64. I've been having to guess this whole time and try for answers on my calculator, because solving for x has not been working. I think I might be doing it wrong...
Second is, I didn't understand this video too well. I don't understand what the graph has 
to do with any of this, even though I've been okay with everything pertaining to logarithms so far. Any answer is much appreciated. Thank you!

Accepted Answer

When the variable is in the exponent, you need to use logarithms of whatever the base of the exponent is.
For 2^x = 1 / 64, the base is 2.  Therefore, we'll be taking log base 2 of each side of the equation.
But before doing that, it's usually easiest to express both sides of the equation using the same base.
So, 2^x = 1 / 64 = 1 / 2^6 = 2^(-6)
log_2 ( 2^x ) = log_2 ( 2^-6 )
x = - 6
Hope this helps.  
(I'll let someone else pick up on your question about the graph.)

Question 6

Hi, can you please explain how did you know that y=log_b(x) is inverse of y=b^x. a detailed answer would be appreciated. Thanks.

Accepted Answer

The actual definition of a logarithm that proves this is several years ahead of the math you're having now, so it wouldn't mean much to you at this stage.
 
But, the short answer is that the logarithm was invented to be the inverse operation of an exponential function. In other words, there is a difficult mathematical process that was worked out over the course of about 80 years by Nicholas Mercator, Leonhard Euler and others. The term "log" is just a shorthand way of expressing this difficult bit of math.
 
So, a log is just a quick way of writing the inverse function of an exponential function. Thus, by definition, the log must be the inverse function of the exponential function.
 
So, the reason you are not given exactly what a log is or how to compute it by hand at this level of study is that it is very difficult math. Since we have calculators that can do this math for us, it is possible for people to use logs without having to master very advanced and difficult computations.

But, just for reference, here is one way to write the actual definition of a logarithm
log_b (x)= lim n→0 [x^(n) - 1] / [b^(n) - 1]
where b is the base of the logarithm.

Question 7

To solve for other points, couldn't I figure out that b=4 from the pattern in y=b^x and go from there?

Accepted Answer

From this set it is obvious that b=4, but the point of the exercise is to highlight the fact that the logarithmic function is an inverse of the exponential function, meaning that you still could have mapped the second set of points simply by reflecting over the line y = x, even if the values present did not provide a simple/obvious value for b.

Question 8

My book has the following example:
You can choose a prize of either a $20,000 car or one penney on the first day, double that (2cents) on the 2nd day, and so on for a month. On what day would you receive more than the value of the car?
$20,000 is 2,000,000 cents. One day you would receive 1 cent, or 2^0 cents. On day 2, you would receive 2 cents or 2^1, and so on. So, on day n you would receive 2^(n-1) cents.
Here is the solution:
2^(n-1)>2X10^6
log 2^(n-1)>log(2X10^6)
(n-1 log 2>log 2+log10^6
(n-1)log2>log2 +6
n-1>(log2+6)/log2
n>(0.301+6)/(0.301)+1
n>21.93
The answer is : On day 22

Now based on the problem above the book asks:
Suppose that you receive triple the amount each day. On what day would you receive at least 1 millon dollars?

I solved it the following way but I could not get the right answer which is day 18.
3^(n-1)>10^6
log 2(n-1)>log10^6
(n-1)log3 >lo 10^6
(n-1)log3>6
n-1 >6/log3
n >6/log3 +1
I got 13.58, but the book says 18 days
I appreciate your help. Thanks

Accepted Answer

The problem has to do with pennies as being .01, so 1 million dollars should be 10^6 dollars or 10^8 pennies.  Try that to see if it works correctly.

Question 9

I've seen a notation such as 10 log (25/5) = 10 log (5).  I'm confused about the "10" and the "log" placement.  Is this the same as saying log_10 5?  Another (very similar) notation is 20 log (0.1/2)=20 log (0.05).  Again, why put the 10 and the 20 before the "log"?  Thank You!

Accepted Answer

Whenever there's no base shown after "log", it means that the base is *10*. A number before log simply means multiplication. Your example:
```
20 log (0.1/2)
``` is the equivalent of saying:
```
20 * log_10 (0.1/2)
```

Algebra 2 (Eureka Math/EngageNY)

Course: Algebra 2 (Eureka Math/EngageNY) > Unit 3

Relationship between exponentials & logarithms: graphs

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