Log 6= 0.778 ( base 10),
In other words when we multiply 10 by itself 0.778 times we get 6.
How would I multiply 10 by itself 0.778 times 🤔?
In: Mathematics
A simple property of exponents is that fractional exponents are roots. So the square root of 10 is really 10^(1/2) aka 10^0.5. This is how exponents work and doesn’t require any understand of logarithms to apply this.
It’s not an exact meaning to “multiplying by itself”
Clearly you can’t multiply 10 by itself by point whatever times.
But you can fill in the gaps with a different math function. (In this case exponentiation). As long as the exponent is a whole number it’s the same as multiplying the number by itself. But if it’s a decimal, it just isn’t the same as repeated multiplication and can only be understood as exponentiation.
So yeah, the answer is 10^0.778
The square root of 10 is about 3.16227766, and that number is like 1/2 of multiplying by 10. (In other words, multiplying by 3.16227766… twice is exactly like multiplying by 10.)
The cube root (third root) of 10 is about 2.15443469, so that’s like a third of multiplying by 10. (Multiplying by 2.15443469… three times is the same as multiplying by 10.)
The fourth root of 10 is 1.77827941… and that’s like a fourth of multiplying by 10.
And so on. There are an infinite number of roots of 10, and each of them is like multiplying by ten 1/2, 1/3, 1/4 or whatever number of times. If you multiplied together some combination of those roots (for instance, the 10th root 7 times, the 100th root 7 times and the 1000th root 8 times) that would be equivalent to multiplying by 10, 0.778 times.
>How would I multiply 10 by itself 0.778 times 🤔?
You wouldn’t.
But if you multiplied 10 by itself 778 times, you’d get (approximately, since log_10 (6) isn’t exactly 0.778) the same result as if you multiplied 6 by itself one thousand times.
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Let’s try this again.
>How would I multiply 10 by itself 0.778 times 🤔?
You’d find a number N, that, if you multiplied it by itself 1,000 times, would give you 10. Then you multiply N by itself 778 times.
So the core of exponentials is this idea that your *growth* is proportional to your *population* or your current value.
Populations grow exponentially because each person can make multiple new people, who can go on to make multiple new people.
They don’t add in a sort of “plus plus” kind of way
They add together in a “times times” kind of way.
When you count in a “plus plus” kind of way, then 7/10 means “70% of 1”, because you are 70% over on the horizontal, and 70% up on the vertical.
But when you count in a “times times” kind of way, you don’t get that linear symmetry going on.
For example
If lillypads double in number every day
and on day 100 there are 100 lillypads
then on which day were there only 50 lillypads?
The answer is day 99
Because the next day it doubles to 100.
So now it takes some 90% of the time to get only 50% of the way, and the last 50% is covered by the last 10% of time.
This is the core of exponential growth, and logarithms are the exponential inverse of exponential growth.
So when you see that log 6 = .778
what this is telling you is that to reach 60% (out of 100%) requires 77.8% of the time (out of 100%)
Another example for the above “1” case
Log(12) of base 10
Is asking the question: “If I want to travel 120% of the distance (12 over 10), how much time will it take?”.
The calculation produces 1.08 which signifies that you traveled 120% of the distance in 108% of the time.
What is 10^(1/2)?
For positive integers x, y we have 10^(x) * 10^(y) = 10^(x+y). Example: 10^(3) * 10^(2) = (10*10*10) * (10*10) = 10^(5).
Now suppose we want the same property to hold for real values of x and y too, because that would be nice.
So, if x = 10^(1/2), what must be the value of x?
We have x*x = 10^(1/2) * 10^(1/2) = 10^(2*(1/2)) = 10^(1) = 10. Thus, 10^(1/2) must be the square root of 10.
Similarly, y = 10^(0.778) = 10^(778/1000) is the unique positive real number such that y^(1000) = 10^(778).
Well, I want to try to help you tackle this is two ways.
First, you have to know that the nth root of x^m is x^(m/n). Now, since 0.778 = 778/1000, then 10^0.778 = 10^(778/1000) = the 1000th root of 10^788.
These are properties of exponents that should should get very used to:
x^(m) * x^(n) = x^(m+n)
x^(m) / x^(n) = x^(m-n)
(x^(m))^(n) = x^(m*n)
(x*y)^(m) = x^(m)*y^(m)
x^(-n) = 1 / x^(n)
x^(0) = 1
Once you have those firmly understood, then helping you understand what a logarithm is is much simpler. Let’s try. Follow along and answer these if you can at home. (You’ll learn it much faster than if you’re given the answer.)
What power do you raise 3 to in order to get 27? [Hint: I am asking you for y in: 3^(y)=27]
What power do you raise 4 to in order to get 16?
What power do you raise 5 to in order to get 625?
… [This question is such an easy one to ask that it gets asked a lot. Let’s try to generalize it.]
What power do you raise **b** to in order to get **x**?
If you answered the first three, then you were already doing logarithms. Logarithms are asking you what power of **your base** you have to raise it to get **a specific number, x.** [In an equation b^(y)=x]
Hence was born log_b(x). Everyone happens to use a base 10 number system, so oftentimes when you see log(x) it is implied that b = 10. There are some scenarios where it is assumed that b = e = 2.7182818284590452…, but it general that is referred to as ln(x).
Simple statements like that are often just the first step in a generalized description: enough to give you an idea of a concept, but might not tell the whole story.
In this case, think of how exponents work. 2·2 = 2^(2) = 4. 2·2·2 = 2^(3) = 8. 2^(5) = 2·2·2·2·2 = 32, which is also 4·8 = 2^(2)·2^(3). Something like this works for every base number and any exponent, so we can generalize that to x^(a)·x^(b) = x^(a+b): to multiply powers of the same base, you can just add their exponents together.
Now look at roots. By definition, √x · √x = x. If we write that as exponentials, we get (√x)^(2) = x^(1). It makes sense then to consider a square root the halfth power since 2·½ = 1: √x = x^(½) = x^(0.5). Higher-power roots are then just fractional exponents.
And now back to logarithms. 0.778 is the same as 778/1000, so 10^(0.778) = 10^(778/1000) = ^(1000)√(10^(788)). You probably don’t want to calculate the exact value that way, but that’s what it means.
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