Finding the Value of axb on the Unit Circle | Round to the Nearest Thousandths

AI Thread Summary
The problem involves finding the value of axb given that the point (log a, log b) lies on the unit circle, which is defined by the equation x² + y² = 1. The initial attempts to solve the equation led to confusion regarding the correct interpretation of logarithmic identities. It was clarified that the correct formulation is (loga)² + (logb)² = 1, and that the values of a and b can vary significantly, leading to multiple potential solutions. The discussion emphasizes the importance of correctly applying logarithmic properties and confirms that the base of the logarithm does not affect the overall relationship. Ultimately, the problem illustrates the complexity of finding specific values for a and b under these conditions.
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Homework Statement


the point (log a, log b) exists on the unit circle. find the value of axb. round to the nearest thousandths.

Homework Equations


x2 + y2 = 1

The Attempt at a Solution



x2+y2 = 1
loga2+logb2 =1
2loga+2logb = 1
2(loga+logb) = 1
loga + log b = 0.5
logb = 0.5−loga
now i try and subsitute logb in
loga2+(0.5−loga)2 = 1

when i did this it wouldn't work after the last step. so this is what i tried next

(loga)2 + (log b)2 = 1
loga = √1-(logb)2

then i did

(√1-(logb)2)2 + (logb)2 = 1
1-(logb)2 = 1-(log b)4
-(logb)2 + (logb)4 = 0
factored out (logb)2 so i got
1 = logb
and 101 = b so b = 10 but I am not sure if that's right either because it seems iffy to me
i think i was on the right track with the first one but idk can someone help please
 
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dfklajsdfald said:
the point (log a, log b) exists on the unit circle. find the value of axb. round to the nearest thousandths.
That makes no sense. There are infinitely many solutions.
E.g. a=1, b=e gives e;
ab=1 gives 1;
a=b gives e√2.
Please check you have stated the question exactly.
 
nope i just checked the question and what i said is exactly what's written. it says: the point (log a, log b) exists on the unit circle. find the value of a times b. round to the nearest thousandth
 
dfklajsdfald said:
nope i just checked the question and what i said is exactly what's written. it says: the point (log a, log b) exists on the unit circle. find the value of a times b. round to the nearest thousandth
I just noticed you were assuming log base 10, whereas I assumed natural logs, but my answer applies whatever the base. It is more obvious if we substitute x=log a, y= log b, v=log (ab). Let the base be c. ab=cxcy=cx+y, so v=x+y. So it is the same as asking for the value of x+y given that x2+y2=1.
 
dfklajsdfald said:

Homework Statement


the point (log a, log b) exists on the unit circle. find the value of axb. round to the nearest thousandths.

Homework Equations


x2 + y2 = 1

The Attempt at a Solution



x2+y2 = 1
loga2+logb2 =1
2loga+2logb = 1
No.
The second equation is ##(\log(a))^2 + (\log(b))^2 = 1##. This is not the same as ##\log(a^2) + \log(b)^2 = 1##.
dfklajsdfald said:
2(loga+logb) = 1
loga + log b = 0.5
logb = 0.5−loga
now i try and subsitute logb in
loga2+(0.5−loga)2 = 1

when i did this it wouldn't work after the last step. so this is what i tried next

(loga)2 + (log b)2 = 1
loga = √1-(logb)2

then i did

(√1-(logb)2)2 + (logb)2 = 1
1-(logb)2 = 1-(log b)4
-(logb)2 + (logb)4 = 0
factored out (logb)2 so i got
1 = logb
and 101 = b so b = 10 but I am not sure if that's right either because it seems iffy to me
i think i was on the right track with the first one but idk can someone help please
 
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