Problem 29 - 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. $$\log \sqrt{100 x}$$

Step-by-Step Solution

Answer

The expanded form of the expression \(\log \sqrt{100x}\) is \(1 + \frac{1}{2} \log x\).

Step 1: Apply the property of logarithms for roots

The general property of logarithms for roots is \(\log_a\sqrt[b]{x} = \frac{1}{b} \log_a x\). Applying this property to our problem, the expression \(\log \sqrt{100x}\) becomes \(\frac{1}{2} \log(100x)\).

Step 2: Apply the property of logarithms for multiplication

The property of logarithms for multiplication states that \(\log_a(xy) = \log_a x + \log_a y\). Let us apply this to our problem. We obtain \(\frac{1}{2} (\log 100 + \log x)\).

Step 3: Evaluate the logarithm of 100 (if possible)

The logarithm base 10 of 100 is just 2, because \(10^2 = 100\). This, our expression becomes \(\frac{1}{2} (2+ \log x)\).

Step 4: Simplify the expression

Finally, by simplifying the last expression, we obtain \(\frac{1}{2} *2+ \frac{1}{2}* \log x\), which equals to \(1+ \frac{1}{2} \log x\).