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 logarithmic expression is \(\frac{1}{2} (1 + \ln x)\).

Step 1: Rewrite the Square Root

Start by rewriting the square root in the logarithm. The square root of a number is the same as raising that number to the power of 1/2. So, \(\ln \sqrt{e x}\) = \(\ln (e x)^{1/2}\).

Step 2: Apply the Logarithm Power Rule

Use the logarithm property that allows to take an exponent out of the log and make it a multiplier on the log. That is, \(\ln a^n = n \ln a\). So, \(\ln (e x)^{1/2}\) can be written as \(\frac{1}{2} \ln (e x)\).

Step 3: Apply the Logarithm Product Rule

Now apply the property of logarithms that the log of a product is the sum of the logs. This means \(\ln ab = \ln a + \ln b\). So, \(\ln (e x)\) turns into \(\ln e + \ln x\). Substitute this back into the expression from Step 2, resulting in \(\frac{1}{2} (\ln e + \ln x)\) .

Step 4: Simplify the Logarithm of e

The natural logarithm's base is e, so \(\ln e\) is 1. To simplify, replace \(\ln e\) with 1 in the expression from Step 3, ending up with \(\frac{1}{2} (1 + \ln x)\).

Key Concepts

Logarithm Power Rule

The logarithm power rule is a handy tool when dealing with exponents inside a logarithm. This rule states that if you have a logarithm of a number with an exponent, you can move the exponent to the front as a multiplier. It's written as \( \ln(a^n) = n \ln(a) \). For instance, if you encounter an expression like \( \ln (x^3) \), you can simplify it by bringing down the exponent: \( 3 \ln(x) \).

• This property makes complex expressions simpler to handle.
• It transforms the task of working with powers into a straightforward multiplication.

In our original problem, we used this rule by rewriting the square root as a power of 1/2, turning \( \ln \sqrt{ex} \) into \( \ln (ex)^{1/2} \). Then, we applied the power rule to get \( \frac{1}{2} \ln (ex) \). Remember, using the power rule with natural logarithms still simplifies just like with common logarithms.

Logarithm Product Rule

The logarithm product rule allows us to break down a logarithm of a product into a sum of logarithms. If you have a logarithm of two numbers multiplying, you can separate them. This is captured by the rule \( \ln(ab) = \ln(a) + \ln(b) \).

• This rule is useful because it turns the multiplication inside a log into simple addition.
• It is particularly handy when dealing with products of variables or numbers.

In the exercise provided, after simplifying using the power rule, we apply the product rule. We had \( \ln(ex) \), which, by using the product rule, becomes \( \ln e + \ln x \). This simplifies the expression by separating it into individual components. The product rule is a key property when expanding functions or simplifying complex logs.

Natural Logarithm

Natural logarithms, often denoted as \( \ln \), are special kinds of logarithms where the base is the mathematical constant \( e \), approximately equal to 2.718. They are called "natural" because \( e \) frequently appears in various mathematical contexts, especially in calculus and complex numbers.

• The natural logarithm has properties similar to common logarithms, such as the power and product rules.
• One defining property of the natural logarithm is \( \ln e = 1 \), because any base raised to its first power equals itself.

In the exercise, after expanding and applying the logarithm properties, \( \ln e \) was simplified to 1, as the base is the same as the argument, simplifying our result further to \( \frac{1}{2}(1 + \ln x) \). Understanding the natural logarithm aids in recognizing when these simplifications can occur, making it easier to manage complicated expressions.