if loga base b*loga base c+logb base a*logb base c+logc base a*logc base b=3 then find the value of abc

To solve the equation given by loga base b * loga base c + logb base a * logb base c + logc base a * logc base b = 3, we can start by simplifying the logarithmic expressions using the change of base formula. The change of base formula states that logx base y = logy base x / logy base y. We can express the logarithms in a more manageable form.

Breaking Down the Components

Let's denote:

• x = loga base b
• y = logb base c
• z = logc base a

Using these variables, we can rewrite the original equation as:

x * (1/y) + (1/x) * y + z * (1/z) = 3

Substituting Values

Now we can rewrite the terms:

• xy + yz + zx = 3

Using the Property of Logarithms

From the properties of logarithms, we know:

• xyz = loga base b * logb base c * logc base a = loga base a = 1

Finding Relationships

This implies:

• xyz = 1

Now, we can use this relationship to substitute back into our equation:

Formulating a Polynomial

We can also express the equation in terms of x, y, and z:

x + y + z = 3

We can use the fact that if we set:

t = x + y + z

Then we also know:

• xyz = 1

Exploring Roots

From the relationships, we form a polynomial where x, y, and z are roots:

t^3 - (x + y + z)t^2 + (xy + xz + yz)t - xyz = 0

Substituting the known values, we get:

t^3 - 3t^2 + (3)t - 1 = 0

Solving the Polynomial

Using the Rational Root Theorem or synthetic division, we can find the roots of the polynomial, which will give us the values of x, y, and z. However, since we are interested in finding abc, we can shortcut to the result:

Final Calculation

Recall that:

• abc = a^(logb base a) * b^(logc base b) * c^(loga base c)

Since we established that xyz = 1 and x + y + z = 3, we can conclude that:

abc = 10^(3) = 1000.

Thus, the value of abc is 1000.