If (p-1)(q-1)(r-1) ≠0, then the value of the determinant

If $(\mathrm{p}-1)(\mathrm{q}-1)(\mathrm{r}-1) \neq 0$, then the value of the determinant $\left|\begin{array}{lll}\log _{\mathrm{P}}\left(\frac{\mathrm{px}}{\mathrm{y}}\right) & \log _{\mathrm{q}}\left(\frac{\mathrm{y}}{\mathrm{z}}\right) & \log _{\mathrm{r}}\left(\frac{\mathrm{z}}{\mathrm{x}}\right) \\ \mathrm{q}, \mathrm{r} \text { are positive }) \text { is } & \log _{\mathrm{p}}\left(\frac{\mathrm{y}}{\mathrm{z}}\right) & \log _{\mathrm{q}}\left(\frac{\mathrm{q} z}{\mathrm{x}}\right) & \log _{\mathrm{r}}\left(\frac{\mathrm{x}}{\mathrm{ry}}\right) \\ \log _{\mathrm{p}}\left(\frac{\mathrm{px}}{\mathrm{z}}\right) & \log _{\mathrm{q}}\left(\frac{\mathrm{qy}}{\mathrm{x}}\right) & \log _{t}\left(\frac{\mathrm{z}}{\mathrm{ry}}\right)\end{array}\right|$, (where $\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{p}$, (a) 0 (b) 1 (c) $-1$ (d) $\log _{\text {pqr }}(x y z)$

If $(\mathrm{p}-1)(\mathrm{q}-1)(\mathrm{r}-1) \neq 0$, then the value of the determinant $\left|\begin{array}{lll}\log _{\mathrm{P}}\left(\frac{\mathrm{px}}{\mathrm{y}}\right) & \log _{\mathrm{q}}\left(\frac{\mathrm{y}}{\mathrm{z}}\right) & \log _{\mathrm{r}}\left(\frac{\mathrm{z}}{\mathrm{x}}\right) \\ \mathrm{q}, \mathrm{r} \text { are positive }) \text { is } & \log _{\mathrm{p}}\left(\frac{\mathrm{y}}{\mathrm{z}}\right) & \log _{\mathrm{q}}\left(\frac{\mathrm{q} z}{\mathrm{x}}\right) & \log _{\mathrm{r}}\left(\frac{\mathrm{x}}{\mathrm{ry}}\right) \\ \log _{\mathrm{p}}\left(\frac{\mathrm{px}}{\mathrm{z}}\right) & \log _{\mathrm{q}}\left(\frac{\mathrm{qy}}{\mathrm{x}}\right) & \log _{t}\left(\frac{\mathrm{z}}{\mathrm{ry}}\right)\end{array}\right|$, (where $\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{p}$,
(a) 0
(b) 1
(c) $-1$
(d) $\log _{\text {pqr }}(x y z)$

Key Concepts

Change of Base Formula

The change of base formula is a key concept in logarithms that enables the conversion of a logarithm from one base to another. This formula, log_b(x) = log_k(x)/log_k(b), where k is any positive number not equal to 1, is particularly useful when expressions involve logarithms with different bases. It helps in unifying the expressions to a common base, thereby simplifying the algebraic manipulation required in solving problems.

Determinant Evaluation in Matrix Algebra

Determinants provide a scalar value that encapsulates important properties of a matrix, including whether it is invertible and its scaling effect on volume. The evaluation of determinants, especially in matrices with complex entries, often involves strategic row or column operations and the application of standard formulas such as expansion by minors. Understanding these properties is crucial when simplifying determinants that include non-standard elements like logarithmic expressions.

Logarithm Properties

Logarithms transform multiplicative relationships into additive ones through identities such as the product, quotient, and power rules. These identities are fundamental in simplifying and comparing expressions, especially when they involve complex ratios or products. They allow one to break down or combine logarithmic expressions, which is crucial when dealing with equations or matrices that contain multiple logarithms.

Transcript

00:01 So in the given question we have a determinant we have a determinant that is given us log px divided by y log of base p of p x by y log with base q of y of z and log with base r of z by x so this is the first row of the determinant in the second row we have log with base p of y by z log with base q of q z by x and log with base r of x x and log with base r of x by r times y.

01:10 Next we have the third row in which we are told log with base p of px by z, log with base q of qy by x and we have log with base r of z divided by r.

01:40 So now we are told that if p minus 1 times q minus 1 times r minus 1 is not equal to 0, then what value does this a determinant have? so we should evaluate the given determinant, right? so we should simplify this determine doing a few row operations.

02:07 So what we should know before doing a row, the row operations involving this logarithm is that we should know a few logarithmic properties, right? we should know what would we have when we multiply them or add them and so on.

02:25 So the property of a logarithm, that is when we add log of a plus log of b, we can write it as log of we can write it as log of a times b right and similarly when we take log of a minus log of b we can write it as log of a divided by b so this is what we need to keep in mind while doing row operations on when doing additional subtraction between logarithms right we can simplify the result as log of a times b and log of a divided by b in different cases so what we are going to do is let's simplify this row algorithm by taking the row operation r1 changes to r1 plus r2 minus r3 right so let's take each element of this operation separately and after finding what this element, after finding what the element is after doing the raw operation, we can write the determinant, right? so in row 1 we have log of log with base p of p x by y plus in row 2 we have log with base p of px by y, of y by z and minus log with base p of in r3 we have px by z right so first let's do the addition operation on these two logarithms and we would have log with base p of px by y times y by z minus log with base p of p x by z right so this would simplify to we would have we can cancel y and y from the numerator and denominator so we would have log with base p of p x divided by z minus log with base p of p x divided by z which means it is equal to 0 right so we got the first element of the of the first row as 0 similarly let's check the second element of the first row of the second element of the first row so after the row operation what we would have is we have log q log with base q of y by z plus log with base q of q divided by x minus log with base of q y divided by x so base q of y divided by x so we would have is let's do the addition operation over here first so this is the second element right we just found the first element now the second element in the first row so we can write log with base q of y by z times qz by x the z and z can be cancelled off from the numerator and denominator so we will have log with base q of q y divided by x minus log with base q of qy divided by x so this is again equal to zero since they are the same logarithms right it is log with base q of qy by x minus log with base q of qy by x minus log with base q of qy by x now now lastly we have the third element in the row and this is equal to log with base r of z by x plus log with base r of x by r of x by r minus log with base r of z divided by r.

07:31 So this is what we have and on doing the addition operation we would have log with base r of z by x times x by r...

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