Final answer:
To combine the expression 4 log x - 6 log (x+2) into a single logarithm, we use the exponentiation and division rules for logarithms to obtain log((x^4)/(x+2)^6).
Step-by-step explanation:
To write the expression 4 log x - 6 log (x+2) as a single logarithm, we can apply log rules.
The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number: log(x^n) = n log x.
The logarithm of a product of two numbers is the sum of the logarithms of the two numbers: log xy = log x + log y.
The logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers: log(x/y) = log x - log y.
Using these rules, we can
rewrite the given expression:
4 log x - 6 log (x+2) can be rewritten as log(x^4) - log((x+2)^6), which simplifies to log((x^4)/(x+2)^6) according to the division rule for logarithms.