Which expression is equivalent to $\log 3(x+4)$ ?
The objective is to identify an equivalent expression for the logarithmic expression \(\log(3(x+4))\). Logarithms follow specific algebraic properties that allow us to expand or contract logarithmic expressions. One of the fundamental rules is the product rule of logarithms.
The product rule of logarithms states that for any positive real numbers \(M\) and \(N\), and a base \(b > 0\) (where \(b
eq 1\)), the logarithm of a product is equal to the sum of the logarithms of the factors:
\[\log_b(MN) = \log_b(M) + \log_b(N).\]
In our given expression, \(\log(3(x+4))\), the base is implicitly \(10\). We can identify the two factors as \(M = 3\) and \(N = x+4\).
Using the identified factors, we substitute them into the product rule formula:
\[\log(3(x+4)) = \log 3 + \log(x+4).\]
This result directly matches the third option provided. The other options involve incorrect transformations, such as using the quotient rule or an incorrect expansion of the product. Thus, the correct equivalent expression is \(\log 3 + \log (x+4)\).
Logarithms are used in various fields, such as chemistry to measure the acidity of a solution (pH scale) and in finance to calculate interest over time. For example, if you are calculating the growth of an investment where the total return is a product of several factors, you can use logarithmic properties to simplify these complex multiplications into simple additions, making the calculations much easier to handle and visualize. This is similar to how we expanded \(\log(3(x+4))\) into \(\log 3 + \log(x+4)\).