If log p = x and log q = y , evaluate the following in terms of x and y:

Question

1. log (p^-9q^-9)

2.log sqrt(p^3q^4)

3.log p^-5/q^1

4.logp^7/logq^8

5.(log p^-5)^-9

Answer 1

You mostly need to use logarithm power and product rules to rewrite these problems in terms of log(p) = x and log(q) = y. For part 2, you need to distribute the square root across the powers of p and q and then adjust the terms to be log(p) = x and log(q) = y. Part 3 uses the logarithm difference rule. Part 4 only uses the power rule.

1. log(p^-9q^-9)

log(p^-9) + log(q^-9)

-9log(p) - 9log(q)

-9x - 9y

2. log(√p³q⁴)

log(p^3/2q²)

log(p^3/2) + log(q²)

(3/2)log(p) + 2log(q)

(3/2)x + 2y

3. log(p^-5/q¹)

log(p^-5) - log(q)

-5log(p) - log(q)

-5x - y

4. log(p⁷)/log(q⁸)

7log(p)/8log(q)

7x/8y

5. (log(p⁵))^-9

(5log(p))^-9

1/(5x)⁹