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

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Homework Help Overview

The problem involves finding the value of \( a \times b \) given that the point \( (\log a, \log b) \) lies on the unit circle, defined by the equation \( x^2 + y^2 = 1 \). The context is logarithmic relationships and their implications on the unit circle.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the implications of the logarithmic equations derived from the unit circle equation. There are attempts to manipulate the equations to express \( a \) and \( b \) in terms of each other. Some participants express confusion about the assumptions regarding the logarithm's base and the nature of the solutions.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some have pointed out potential ambiguities in the question, while others are attempting to clarify their reasoning and the mathematical steps involved. There is no explicit consensus on the approach or the correctness of the interpretations yet.

Contextual Notes

There is a noted assumption about the base of the logarithm, which may affect the interpretation of the problem. Additionally, some participants highlight that the problem may have infinitely many solutions based on different values of \( a \) and \( b \).

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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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