As always with my math questions - not homework; just trying to learn me some math.
Can someone explain why this property of logarithms works:
Take 2^(log(2)8). Log (2)8 = 3, so 2^3 = 8. Simple case.
But using a calculator, I see that 8^(log(8)17) = 8^1.362487+several more decimal places = 17.
I know I should be able to work out the proof, but it is eluding me. Would someone please be so kind as to show it to me.
Well, isn’t it just a case of x^y and logx y being inverse functions? So, combining the two gives the identity function (doesn’t change the value).
This is the definition of the logarithm function.
log(x) y = z (Log to the base of x of y is z) means that z^x = y.
If you want an algebraic “proof” take
b^(log_b(a))=a
Take the log_a of both sides, use the change of base to convert all logs to natural logs. You’ll end up with 1=1. (For b,a =/= 1)
Is there a typo here? log(2)8=3, but 2^3=8, which would be x^z=y. I’m not being nitpicky - I’m just so slow when it comes to these things I want to make sure I have it right.
Jragon I don’t know natural logs yet.
CanadjanI don’t doubt it, but I’m not completely sure what that means.
I suppose you made a calculator mistake.
Note that the calculator probably has no log(8) nor a log(x) y button… how to do it ?
log(8) 17 = log(x) 17 / log(x) 8
So the calculator probably has log (10) … and maybe log(e)… you can use either… just as long as its the same log in use.
You of course need a x^y button or function on the calculator…
Maybe do it step by step in excel so that you can see each step.
Alright, thank you everyone! Taking everything that was said, spending some time with paper and pencil, and I see it now. :smack:
Each log function has a particular “base”, right? So the base-2 log of 8 is 3, the base-10 log of 100 is 2, the base-5 log of 625 is 4, and so forth. (These are all related, since they’re all multiples of each other).
The most commonly used bases are base-10, because of the decimal system, occasionally base-2 (because of doubling), and importantly, the natural log function uses base e. This is an irrational number, roughly around 2.71828, which turns out to have a lot of important properties in calculus, analysis and statistics. It’s the limiting value of the expression (1 + 1/n)^n, when n becomes infinitely large.
For further reading, there’s the Better Explained blog:An Intuitive Guide To Exponential Functions & e
Demystifying the Natural Logarithm (ln)
How To Think With Exponents And Logarithms
Using Logarithms in the Real World
Base-2 logs also have use in computer science, because log[sub]2/sub is the number of bits needed to store n different values.
And my preferred explanation of the significance of e is that it’s the base for which e^x is approximately equal to 1+x, for small x. Or equivalently, ln(x) is approximately equal to x-1, for x close to 1. It’s really the same reason why we prefer using radians (based on pi) to measure angles: Because it makes the slopes nice.
Right. The way I had it explained to me in high school was, it’s the particular exponential function that has a slope of 1 when it passes through the y-axis (because, as you point out, when x = 0, e^x is approximately 1 + x).
Or as you learn later in calculus, at every value of x, the slope of the function e^x is equal to the value of the function itself. (Whereas, for the function 2^x, for example, the slope of the function is equal to about 0.69 times the value of the function, and for 10^x, the slope of the function is equal to about 2.30 times the value of the function.)
And of course, those two numbers are ln(2) and ln(10). And if you get right down to it, the slope of e^x is ln(e) times the value of the function, where ln(e) is of course 1.