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

Step-by-Step Solution

Answer

\(4 + \log x\)

Step 1: Separate Logarithm of Multiple Factors

Use the property of logarithms that states the logarithm of a product is the sum of the logarithms of its factors to rewrite the original logarithm: \[\log (10,000 ⋅ x) = \log 10,000 + \log x\]

Step 2: Evaluate Common Logarithm

Evaluate the logarithm \(\log 10,000\) using the rule of common logarithms, which states that \(\log 10^n = n\)\[\log 10,000 = 4\] Substitute the result back into the expression from Step 1:\[4 + \log x\]

Step 3: Simplified Expression

Finally, we arrive at the expanded and simplified logarithmic expression:\[4 + \log x\] This is as simple as the logarithmic expression can be without any information on the value of \(x\).