Problem 1 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{5}(7 \cdot 3) $$

Step-by-Step Solution

Answer

The expanded form of \(\log _{5}(7 * 3)\) using properties of logarithms is: \(\log _{5}(7) + \log _{5}(3)\).

Step 1: Recognize Which Property to Use

In this case, we're dealing with a logarithm of a product of numbers, \(7 * 3\). We can transform this into the sum of the logarithms of each number by using the following property of logarithms: \(log_b(MN) = log_b(M) + log_b(N)\). We're going to do this for the logarithm base 5.

Step 2: Apply the Property of Logarithms

So, applying the property to our expression results in the following: \(\log_{5}(7*3) = \log_{5}(7) + \log_{5}(3)\).

Step 3: Final Expression

Therefore, we can say that the given logarithm expanded using the property of logarithms is: \(\log_{5}(7*3) = \log_{5}(7) + \log_{5}(3)\). This is already the simplest form of the expression. Please note that it is not possible to compute the actual values without a calculator.