Question

Given log(sqrt(x))8=p and log(y)2=q. write log(8)x^2y in terms of p and q

Answer 1

To rewrite log(8)x^2y in terms of p and q, use the given information and properties of logarithms to find that log(8)x^2y equals 4p + 3q.

Step-by-step explanation:

To express log(8)x^2y in terms of p and q, we need to use the properties of logarithms:

1. The logarithm of a product of two numbers is the sum of the logarithms of the two numbers: loga(xy) = loga(x) + loga(y).
2. The logarithm of a power of a number is the exponent times the logarithm of the number: loga(x^b) = b * loga(x).

Given log(sqrt(x))8=p, we can rewrite it as log(x)8 = 2p, because the square root is equivalent to raising x to the 1/2 power. Similarly, given log(y)2=q, we can use the change of base formula to write log8(y) = q / log8(2). Since log8(2) is equal to 1/3 (because 2^3 = 8), we have log8(y) = 3q.

Now, using these new expressions, we can find log(8)x^2y:

log(8)x^2y = log(8)x^2 + log(8)y by property (1)

Since we have already established that log(x)8 = 2p and log8(y) = 3q, our expression becomes:

log(8)x^2y = 2 * 2p + 3q = 4p + 3q