First, to re-write the expression log((ab^(3))/(c^(2))), we can use the properties of logarithms. These properties are foundational rules in logarithm operations.
Looking at the given expression, we see it's a logarithm of a quotient. One of the properties of logarithms says that the logarithm of a quotient is equal to the difference of the logarithms. In mathematical forms, this property is expressed as: log(m/n) = log(m) - log(n).
Applying this property to our given equation, we can break down log((ab^(3))/(c^(2))) into two parts:
log(ab^3) and log(c^2), separated by a minus sign because that's what the rule says.
So, it looks like this now:
log(ab^3) - log(c^2).
We can see that both expressions now inside the logarithms are products and there's another property of logarithms stating that the logarithm of a product is the sum of the logarithms. Mathematically, this rule is framed as: log(mn) = log(m) + log(n).
For the first term, we apply this rule and get:
log(a) + log(b^3) - log(c^2).
Again, we notice the presence of powers (or exponents) in the terms. There's another logarithm rule that handles the powers in the expression. It states that the logarithm of a power equals the product of the power times the logarithm of the base. This rule is written as: log(m^n) = n * log(m). We can apply this rule to the terms log(b^3) and log(c^2), which transforms the equation as:
log(a) + 3 * log(b) - 2 * log(c).
So, the equivalent expression for the original log((ab^(3))/(c^(2))) is log(a) + 3 * log(b) - 2 * log(c). This simplifies and reduces the complexity of our original expression using the simple known rules of logarithms, which makes it easier to manipulate or evaluate.