Problem 6 - 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. $$ \log (10,000 x) $$

Step-by-Step Solution

Answer

The expanded form of \(\log (10,000 x)\) is \(4 + \log x\).

Step 1: Break Down the Expression

First, the expression needs to be separated according to the product rule of logarithms, which states that \(\log_b(mn) = \log_b m + \log_b n\). Here, \(m=10000\) and \(n=x\), thus the expression \(\log (10,000x)\) can be rewritten as \(\log 10000 + \log x\).

Step 2: Evaluate the Logarithm

Next, we evaluate the \(\log 10000\) portion of the expression. In the context of common logarithms (base 10), \(\log 10000 = \log 10^4\), and using the rule \(\log_b b^a = a\), we have \(\log 10^4 = 4\). So, the expression \(\log 10000 + \log x\) becomes \(4 + \log x\).

Step 3: Finalize the Expression

Lastly, the unaltered portion of the initial expression, \(\log x\), remains in the final expression as-is because there's no further simplification possible without knowing the value of \(x\). Thus, the final expanded form of the expression is \(4 + \log x\).