Problem 3 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{7}(7 x)$$

Step-by-Step Solution

Answer

The expanded and simplified logarithmic expression is \(1 + \log_{7}(x)\).

Step 1: Identify the properties of logarithms

One of the properties of logarithms is the Product Rule, which stipulates that the log of a product is the sum of the logs. Meaning, \(\log _{b}(mn) = \log _{b}(m) + \log _{b}(n)\). This property will be applied in this exercise.

Step 2: Apply the Product Rule

Use the Product Rule to expand \(\log _{7}(7x)\) into \(\log _{7}(7) + \log _{7}(x)\)

Step 3: Simplify the expression

As per the definition of logarithms, if \(\log_{b}(a) = n\), then \(b^{n} = a\). Therefore, \(\log _{7}(7) = 1\) since \(7^{1}=7\). Substitute this into the expression to obtain \(1 + \log _{7}(x)\)