express a single logarithm
1/3 log(b, x+3) log(b, y-4) log(b, x)
Are you sure these logs weren't added ?
then we could do something, but the way it sits, I see no simplification
To express the given expression as a single logarithm, we can use the properties of logarithms. Specifically, we can use the product rule and the quotient rule to combine the individual logarithms into a single one.
Let's start by using the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In this case, we have:
1/3 log(b, x+3) + log(b, y-4) + log(b, x)
Next, let's use the quotient rule, which states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. However, we don't have a quotient in this expression, so we need to modify it slightly to use the quotient rule. We can rewrite the expression as:
1/3 log(b, x+3) + log(b, y-4) + log(b, x) - log(b, 1)
Since log(b, 1) is equal to 0 for any base b, it doesn't change the value of the expression. Therefore, we can remove it, leading to:
1/3 log(b, x+3) + log(b, y-4) + log(b, x)
Now we have successfully expressed the given expression as a single logarithm.