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

Step-by-Step Solution

Answer

So, the expanded form of the given logarithmic expression is \(1/2 * (1 + \ln(x))\).

Step 1: Identify the properties of logarithms

First, we need to understand the following properties of logarithms: 1) \(\ln(a * b) = \ln(a) + \ln(b)\), this property allows us to separate a multiplication under a logarithm into a sum of logarithms; 2) \(\ln(a^n) = n * \ln(a)\), this property allows us to 'bring down' an exponent in a logarithmic expression.

Step 2: Apply the laws of exponents

Start by recognizing the square root notation as a fractional exponent, i.e., \(\sqrt{e x} = (e x)^{1/2}\). So, our equation now looks like - \(\ln((e x)^{1/2})\).

Step 3: Apply logarithm properties

Using the property \(\ln(a^n) = n * \ln(a)\), we can 'bring down' the exponent 1/2 in front of the logarithm, the expression becomes - \(1/2 * \ln(e x)\). After this, apply the property \(\ln(a * b) = \ln(a) + \ln(b)\), to separate the multiplication under the logarithm into a sum of logarithms, the expression further transforms as - \(1/2 * (\ln(e) + \ln(x))\).

Step 4: Evaluate the logarithm of e

From a property of the natural logarithm -- \(\ln(e)\) equals 1, we can substitute this into the equation to simplify it further into - \(1/2 * (1 + \ln(x))\).