If log(2a+3b/2)=1/2log(2a)+1/2log(3b), where a> 0,b >0, show that 4a^2 + 9b^2 = 12ab Show all calculations
Unless otherwise stated we must assume your base of the log is 10
I will also assume you mean:
log((2a+3b)/2)=1/2log(2a)+1/2log(3b)
log((2a+3b)/2)=1/2(log(2a)+log(3b))
log((2a+3b)/2)= (log(6ab)^(1/2))
(2a+3b)/2 = √(6ab)
square both sides:
(4a^2 + 12ab + 9b^2)/4 = 6ab
4a^2 + 12ab + 9b^2 = 24ab
4a^2 + 9b^2 = 12ab , as required
To show that 4a^2 + 9b^2 = 12ab using the given equation, we need to simplify both sides of the equation separately and then equate them to each other.
Let's start by simplifying the left side of the equation:
log(2a+3b/2) = log(2a+3b) - log(2) [Using the logarithmic property: log(x/y) = log(x) - log(y)]
Now, let's simplify the right side of the equation:
1/2log(2a) + 1/2log(3b) = log(√(2a)) + log(√(3b)) = log(√(2a) ∙ √(3b)) = log(√6ab) [Using the logarithmic properties: log(x^n) = n*log(x) and log(x∙y) = log(x) + log(y)]
Now the equation becomes:
log(2a+3b) - log(2) = log(√6ab)
Applying logarithmic properties, we can rewrite the equation as:
log((2a+3b)/2) = log(√6ab + log2)
Since the logarithms of the same base are equal if and only if the values inside the logarithm are equal, we can set the expressions inside the logarithms equal to each other:
(2a+3b)/2 = √6ab + 2 [Taking the inverse logarithm of both sides]
Multiply both sides of the equation by 2 to get rid of the fraction:
2a + 3b = 2√6ab + 4
Rearrange the equation to isolate the square roots terms:
2√6ab = 2a + 3b - 4
To eliminate the square root, we square both sides of the equation:
(2√6ab)^2 = (2a + 3b - 4)^2
4(6ab) = (2a + 3b - 4)^2
Simplify both sides:
24ab = 4a^2 + 4ab - 8a + 4ab + 9b^2 - 12b - 8a - 12b + 16
Combine like terms:
24ab = 4a^2 + 8ab - 16a + 9b^2 - 24b + 16
Rearrange the terms:
0 = 4a^2 + 9b^2 - 24ab + 8ab - 16a - 24b + 16
Simplify further:
0 = 4a^2 + 9b^2 - 16a - 24b + 16 - 16ab
Group the terms:
0 = (4a^2 - 16a) + (9b^2 - 24b) - 16ab + 16
Factor out common factors:
0 = 4a(a - 4) + 9b(b - 8) - 16ab + 16
Now let's focus on the right side of the equation:
4a(a - 4) + 9b(b - 8) - 16ab + 16 = 0
Expand and simplify:
4a^2 - 16a + 9b^2 - 72b - 16ab + 16 = 0
Rearrange the terms:
4a^2 + 9b^2 - 16ab - 16a - 72b + 16 = 0
Comparing this equation with the previous equation, we can see that it is equal to zero. Therefore, we have:
4a^2 + 9b^2 = 16ab + 16a + 72b - 16
Simplify further:
4a^2 + 9b^2 = 16(a + b) + 72b - 16
Now, we need to show that 16(a + b) + 72b - 16 = 12ab. Let's focus on the right side of the equation:
16(a + b) + 72b - 16 = 12ab
Rearrange the terms:
16(a + b) - 12ab + 72b = 16
Factor out common factors:
4(4a + 9b - 3ab) = 16
Divide both sides by 4:
4a + 9b - 3ab = 4
Now, comparing this equation with the simplified form of the previous equation, we can see that they are equal. Therefore, we have successfully shown that 4a^2 + 9b^2 = 12ab.