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

Step-by-Step Solution

Answer

Expanded form of the expression \(\log \sqrt{100x}\) is \(0.5(2 + \log(x))\).

Step 1: Apply the square root as exponent

Using the power rule of logarithm which states \(\log(n^p) = p\log(n)\), where \(n\) is the number and \(p\) is the power, square root can be written as exponent \(1/2\). Hence, our expression becomes \(\log((100x)^{1/2})\).

Step 2: Application of Power rule

Applying the power rule \(\log(n^p) = p\log(n)\) again, we get the expression \(0.5\log(100x)\).

Step 3: Application of Product Rule

Apply the product rule of logarithm which states \(\log(ab) = \log(a) + \log(b)\) where \(a\) and \(b\) are numbers, to decompose the \(100x\) inside the log. This gives \(0.5(\log(100) + \log(x))\).

Step 4: Evaluation

Evaluate the \(\log(100)\) because its value can be easily calculated. Any log to the base 10 of 100 is 2, because \(10^2 = 100\). The \(\log(x)\) remains as it is because we cannot evaluate it without knowing the value of \(x\). The final expression after evaluation is \(0.5(2 + \log(x))\).