express log (3x^5y^3/z^3) as the sum and difference of logs with no exponents.

To express log(3x^5y^3/z^3) as the sum and difference of logs with no exponents, we need to utilize the logarithmic properties.

First, let's calculate the logarithms of the individual factors:

1. log(3) - Since 3 is a constant, its logarithm is simply log(3).
2. log(x^5) - From the power rule of logarithms, the exponent 5 can be brought down as a coefficient: 5 * log(x).
3. log(y^3) - Similarly, using the power rule with exponent 3 gives us: 3 * log(y).
4. log(z^3) - Again, applying the power rule with exponent 3: 3 * log(z).

Now, we can rewrite the original expression in terms of logs as follows:

log(3x^5y^3/z^3) = log(3) + log(x^5) + log(y^3) - log(z^3)

Since we want to express it as the sum and difference of logs, we can also write it in the form:

log(3) + 5 * log(x) + 3 * log(y) - 3 * log(z)

This is the result we were looking for.