Problem 30 - 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. $$ \ln \sqrt{e x} $$

Step-by-Step Solution

Answer

The expanded form of the given expression is \(\frac{1}{2} \cdot \ln x\).

Step 1: Express the square root using a rational exponent

The square root of a number can be expressed as raising that number to the power of \(\frac{1}{2}\). So, \(\sqrt{e x}\) is equivalent to \((e x)^{\frac{1}{2}}\). Hence the given logarithmic expression becomes \(\ln (e x)^{\frac{1}{2}}.\)

Step 2: Apply the logarithmic property of exponents

According to the logarithmic property of exponents, the exponent in the argument of a logarithm can be brought out as a multiplier. Hence, \(\ln (e x)^{\frac{1}{2}}\) is the same as \(\frac{1}{2} \cdot \ln (e x).\)

Step 3: Expand the natural logarithm of a product

The natural logarithm of a product can be expressed as the sum of the natural logarithms of the factors. So, \(\ln (e x)\) can be expanded to \(\ln e + \ln x\). Hence the expression becomes \(\frac{1}{2}(\ln e + \ln x)\), which further simplifies to \(\frac{1}{2} \cdot \ln x\) because \(\ln e = 1\).