Question

Use the properties of logarithms to expand the logarithm as much as possible. Rewrite the expression as a sum, difference, or product of logs. log(sqrt(x^7 * y^(-12)))

Answer 1

The logarithm of the square root of the product of x to the 7th power and y to the negative 12th power can be expanded using the logarithmic properties of products, quotients, exponents, and square roots to become (7/2) * log x - 6 * log y.

Step-by-step explanation:

To expand the expression log(sqrt(x^7 * y^(-12))) using the properties of logarithms, we need to apply a set of logarithmic identities:

1. The logarithm of a product log(xy) = log x + log y.
2. The logarithm of a quotient log(x/y) = log x - log y.
3. The logarithm of a number raised to an exponent log(x^n) = n * log x.
4. The logarithm of a square root log(sqrt(x)) = (1/2) * log x, which is a specific case of the third property.

Applying these properties step-by-step:

1. Start by applying the square root property:

log(sqrt(x^7 * y^(-12))) = (1/2) * log(x^7 * y^(-12))

2. Apply the product property:

= (1/2) * (log(x^7) + log(y^(-12)))

3. Apply the exponent property:

= (1/2) * (7 * log x + (-12) * log y)

4. Finally, multiply the exponents inside the parenthesis by 1/2:

= (7/2) * log x - 6 * log y

This is the fully expanded logarithmic expression.