Question 1

I felt my brain straining at time stamp 3:18.  It seems in these problems that you just have to recognize roots to solve these logs without a calculator or one is doomed to trial and error.  How does one solve LOG problems, without calculator or prefigured tables, on numbers that really don't come out nicely like LOG base 3 of 13?

Accepted Answer

One doesn't, really. This describes two methods to calculate logs by hand; one is limitedly useful and the other requires calculating square roots (which can also be done by hand).

http://fiziko.bureau42.com/teaching_tidbits/manual_logarithms.pdf
http://fiziko.bureau42.com/teaching_tidbits/manual_square_root.pdf

Question 2

At the first you've said that log 2 to the base 8 will be 1/3 and log 1/8 to the base 2 will be -3,but that's confusing why can't log 2 to base 8 be -3?

Accepted Answer

This might help. A power of:
3 means that we multiply a base by itself 3 times.
-3 means that we divide 1 by the base 3 times
1/3 means that we are looking for the cube root of the base.

In general terms a certain power......
P means that we multiply a base by itself P times.
-P means that we divide 1 by the base P times
1/P means that we are looking for the P root of the base.

Think about it, and notice that......
log(base 8)2=1/3            →     is the logarithmic form
base 8^(1/3)=2              →      is the exponential form
2=cube root of 8
and
log(base 2)1/8=-3           →     is the logarithmic form
base 2^-3=1/8              →      is the exponential form           
1/8=1/(2*2*2) or 1/2/2/2

Also always keep in mind that exponents are practically the opposite of logarithms.
So just remember that.......
log(B)R=P            →            is the logarithmic form
B^P=R                   →           is the exponential form
where:
B=base (the starting number, and if it is just log R then and there is no base B specified then that means our base B is an automatic 10)
R=resulting number after raising the base B by a certain power of P.
P=power

Question 3

I still do not understand how can exponents be fraction...

Accepted Answer

Well, think about when you learned multiplication.  At first you were told that multiplication is just adding a number to itself the indicated number of times.  But, then you learned about multiplying by fractions and decimals -- but how could add an number to itself 2.13 times?
 
Likewise, with exponents, they tell you that it is just multiplying a number by itself that number of times.  But, now you're learning, just like with multiplication, the exponents aren't so limited -- they can be any number at all, just like you can multiply by any number at all.
 
So, now you are just moving beyond where exponents are so simply described and getting into where they are really useful. In real-world, applied mathematics (for example, things like finance and economics) your exponents are rarely integers.

Question 4

What does 0^0 evaluate to? Because 0 raised to anything is zero, But anything raised to zero is 1... I'm really confused... Is it a complex number by any chance?
thanks!

Accepted Answer

Don't listen to the last two comments. 0^0 is undefined because there are two conflicting ways to find a solution to this problem.

Question 5

How do I find a logarithm of a prime number?

Accepted Answer

Did you mean by:
"prime# as for the exponent part"?
"prime# as for the base"?
or " prime # as for the solution set"?

Well, by reasonable sense, I will assume you mean the 3rd condition, which is definitely possible to be written as logarithm. e.g.:
log 9 (1/2)= 3
or
log 125 (1/3) = 5

I hope that's the answer you've been looking for :-)

Question 6

In this video, Sal gave us a problem. He said that 8^x=2. so, how did he get 1/3 as a value for x?

Accepted Answer

8^x = 2
if you break the 8 into (2*2*2)^x = 2 it´s the same as (2^3)^x = 2, you know that tou can take an expoent inside parenthesis and multiply by the outside expoent, thus:
(2)^3*x = 2 or 2^1.
same bases? cancel the 2´s, now you have 3x = 1, x = 1/3

Question 7

Ok, I understand that exponents can be fractions and I understand how Sal evaluated it. So would log(1023) be 3.009? I'm not allowed to use a calculator.

Accepted Answer

To understand the reason why log(1023) equals approximately 3.0099 we have to look at how logarithms work. Saying log(1023) = 3.009 means 10 to the power of 3.009 equals 1023. The ten is known as the base of the logarithm, and when there is no base, the default is 10. 10^3 equals 1000, so it makes sense that to get 1023 you have to put 10 to the base of a number close  to 3. So while you might not be able to figure out exactly what log(1023) is without a calculator, you are able to get a good idea of what it should be.

Question 8

Can somebody please explain what Sal said starting from 0:40? It confuses me.

Accepted Answer

First:  Are you aware that a root, such as a square root or a cube root, is the same thing as a fractional exponent?  Specifically, 
√x = x^(½)
∛x  = x^(⅓)
∜x = x^(¼)
And so on...
Thus log₈ (2) = x, can be rearranged to   8ˣ = 2
So, to find what x is, you have to know what power you have to put 8 to to get 2.   Since 2³ = 8,  then  ∛8 = 2
But ∛8  is the same thing as 8^(⅓).
So, the power you have to raise 8 to in order to get 2 is ⅓.  Thus, x=⅓  whether you write the problem in the log form or in the exponential form.

Question 9

Explain how you get 10^0=1

Accepted Answer

Anything to the zeroth power is equal to 1. This can be proven in multiple ways.
But let us look at an informal proof:
10⁴/10 = 10³ ← *notice that I subtracted 1 from the exponent*
10³/10 = 10² ← *again we subtracted 1 from the exponent*
10²/10 = 10¹ ← *again we subtracted 1 from the exponent*
10¹/10 = 10⁰ ← *again we subtracted 1 from the exponent*
But 10¹/10 is just 10/10, which is obviously equal to 1
Thus, 10⁰ = 1
So, any number to the zeroth power is just another way of saying a number divided by itself, which is always equal to 1 with the single exception of 0/0 (thus 0⁰ is a special case).
 
What 0⁰ equals depends on whom you ask. To date, there has been no rigorous mathematical proof that establishes what it equals. However, there are rather a lot of mathematical theorems that suggest but don't prove that 0⁰ = 1, thus most mathematicians would state that  0⁰ = 1. However, not all agree, some claim 0⁰ is undefined or indeterminate.

Question 10

I think Sal has made a mistake, 8^1/4 is 2 not 8^1/3. 8^1/3 is 2 2/3 which is roughly 2.66666... recurring.

Accepted Answer

Incorrect. There is a difference between ^ and *. You are thinking of 8*1/4, which is 2.
The ^ indicates the exponent. This uses multiplication, but is not multiplication. Normally you're used to seeing ^2, or squared. This would mean you repeat the number 2 times and then multiply, so 8*8 = 64.

Does this only works when you see the whole numbers? NO!
The fraction 1/2 is the inverse of 2 (or 2/1). Therefore the operation is the inverse, so instead of squared, we need square root. For example 9^1/2 = 3.

Now for the 1/3. If it was 3/1, you would say cubed. Therefore since we have the inverse fraction, we instead need the cubed root. This means what number multiplied by itself 3 times results in the number 8?
This answer is 2. 2*2*2=4*2=8. Therefore 8^1/3=2.

Algebra 2

Course: Algebra 2 > Unit 8

Evaluating logarithms (advanced)

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