If p square+q square =11pq show that
1/2(log p+log q)=
Log(p-q/3)
Log(p+q/root 3)
Correct
I'd like to know the answer
To prove that 1/2(log p + log q) = log(p - q/3) and log(p + q/√3), we need to manipulate the given equation and apply logarithmic properties. Let's break it down step by step:
Given: p^2 + q^2 = 11pq
Step 1: Take the logarithm of both sides of the equation. We can use any base for logarithm; let's use base 10 (log10) for simplicity.
log10(p^2 + q^2) = log10(11pq)
Step 2: Applying logarithmic properties, we can convert the equation to the sum of logarithms:
log10(p^2) + log10(q^2) = log10(11pq)
Step 3: Simplify the logarithms using the exponent rule:
2log10(p) + 2log10(q) = log10(11pq)
Step 4: Divide the entire equation by 2:
log10(p) + log10(q) = 1/2 * log10(11pq)
Now we have 1/2(log p + log q) = 1/2 * log(11pq).
To proceed to the second part of the proof, we need to expand the right side of the original equation (p^2 + q^2 = 11pq) using the square of a difference formula:
p^2 + q^2 = (p + q)(p - q)
Step 5: Divide both sides of the equation by (p - q):
(p^2 + q^2)/(p - q) = p + q
Simplifying further:
(p^2 + q^2)/(p - q) = p + q/1
Step 6: Rearrange the equation:
(p^2 + q^2)/(p - q) - q = p + (q - q)
Step 7: Separate the numerator:
(p^2 - 2pq + q^2)/(p - q) = p
Step 8: Using the difference of squares factorization:
[(p - q)^2]/(p - q) = p
Step 9: Cancel out the common term (p - q):
p - q = p
Step 10: Rearrange the equation:
p - p = q
Step 11: Simplify:
0 = q
From Step 11, we obtained q = 0.
Now let's substitute this value into the equation 1/2(log p + log q):
1/2(log p + log 0) = 1/2 * log(11p(0))
Since the logarithm of 0 is undefined, we cannot continue the proof.
Therefore, the statement 1/2(log p + log q) = log(p - q/3) and log(p + q/√3) is only valid if q = 0.
hmmm.
p^2 + q^2 = 11pq
p^2 - 2pq + q^2 = 9pq
p-q = 3√(pq)
(p-q)/3 = √(pq)
log [(p-q)/3] = 1/2 log(pq)
log(p-q) - log3 = 1/2 log(pq)
I don't know what the heck your choices were, but maybe you can figure it out.