Find the value of determinant . log x log y log z log 2x log 2y log 2z log 3x log 3y log 3z?

Question

Answer

Let x denote the required logarithm.

Therefore, log2√3 1728 = x

or, (2√3)x = 1728 = 26 ∙ 33 = 26 ∙ (√3)6

or, (2√3)x = (2√3)6

Therefore, x = 6.

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Answer

Calculating the Determinant
To find the value of the determinant given, we can use the properties of determinants for a 3x3 matrix. The given matrix represents the logarithms of various values, so we will simplify it step by step to find the determinant.

Step 1: Expand the Determinant
We can expand the determinant along the first row. The determinant of a 3x3 matrix is calculated as:
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Step 2: Substitute the Values
Substitute the logarithmic values into the determinant formula:
det(A) = log x(log 2z * log 3y - log 3z * log 2y) - log y(log 2z * log 3x - log 3z * log 2x) + log z(log 2y * log 3x - log 3y * log 2x)

Step 3: Simplify the Expression
Now, simplify the expression by multiplying and subtracting the logarithmic terms to get the final value of the determinant.
By following these steps, you can calculate the determinant of the given matrix involving logarithmic values.

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