SOLUTION: The lengths of the sides of an equilateral triangle are log4(a), log10(b), log25(a+b) where A and B are positive numbers. What is the value of a/b?

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: The lengths of the sides of an equilateral triangle are log4(a), log10(b), log25(a+b) where A and B are positive numbers. What is the value of a/b?      Log On


   

Question 1207302: The lengths of the sides of an equilateral triangle are log4(a), log10(b), log25(a+b) where A and B are positive numbers. What is the value of a/b?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let x be the side length of the equilateral triangle. Then

log%284%2Ca%29=x --> 4%5Ex=a

log%2810%2Cb%29=x --> 10%5Ex=b

log%2825%2Ca%2Bb%29=x --> 25%5Ex=a%2Bb

Now look for a relationship between the three bases 4, 10, and 25: 4%2A25+=+10%5E2. So

%2810%5Ex%29%5E2=b%5E2
100%5Ex=b%5E2

25%5Ex=%28100%2F4%29%5Ex=100%5Ex%2F4%5Ex
a%2Bb=b%5E2%2Fa
a%5E2%2Bab=b%5E2

Treat this as a quadratic equation with a as the variable and solve for a using the quadratic formula.

a%5E2%2Bab-b%5E2=0

a=%28-b%2Bsqrt%28b%5E2%2B4b%5E2%29%29%2F2 (ignore the other solution, since a has to be positive)

a=%28-b%2Bsqrt%285b%5E2%29%29%2F2

a=%28-b%2Bb%2Asqrt%285%29%29%2F2

a=b%28-1%2Bsqrt%285%29%2F2%29

Divide by b to find the value of a/b.

a%2Fb=%28sqrt%285%29-1%29%2F2

ANSWER: a%2Fb=%28sqrt%285%29-1%29%2F2

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NOTE added after seeing the response from tutor @ikleyn...

This provides a good example of how a given problem is open to solving using very different equally good paths.

ALWAYS be open to the possibility of solving any given problem in different ways. Finding a different (and sometimes better) way to do something is how human knowledge increases.

Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.
The lengths of the sides of an equilateral triangle are log4(a), log10(b), log25(a+b)
where A and B are positive numbers. What is the value of a/b?
~~~~~~~~~~~~~~~ 

We are given 

    log%284%2Ca%29 = log%2810%2Cb%29 = log%2825%2C%28a%2Bb%29%29.


Let  k = log%284%2Ca%29 = log%2810%2Cb%29 = log%2825%2C%28a%2Bb%29%29.


It means that 

    4%5Ek = a,       (1)

    10%5Ek = b,      (2)

    25%5Ek = a + b.  (3)


It implies that

    4%5Ek + 10%5Ek = 25%5Ek.


Divide both sides (all the terms) by  25%5Ek.  You will get

    %284%2F25%29%5Ek + %2810%2F25%29%5Ek = 1,

or

    %282%2F5%29%5E%282k%29 + %282%2F5%29%5Ek = 1.    (4)


Let x = %282%2F5%29%5Ek.  Then equation (4) takes the form

    x%5E2 + x = 1,

or

    x%5E2 + x - 1 = 0.


Its roots are  x%5B1%5D = %28-1+%2B+sqrt%285%29%29%2F2,  x%5B2%5D = %28-1+-+sqrt%285%29%29%2F2.


Our value of x is positive %282%2F5%29%5Ek;  so, we consider only positive root  x = %28sqrt%285%29-1%29%2F2.


Thus we have

    %282%2F5%29%5Ek = %28sqrt%285%29-1%29%2F2,

or

    %284%2F10%29%5Ek = %28sqrt%285%29-1%29%2F2.


But from (1) and (2),  a%2Fb = %284%2F10%29%5Ek.


Thus we proved that  a%2Fb = %28sqrt%285%29-1%29%2F2.


ANSWER.  a%2Fb = %28sqrt%285%29-1%29%2F2 = 0.618033989  (approximately).

Solved.