Problem 20 - 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. $$ \ln \sqrt[2]{x} $$

Step-by-Step Solution

Answer

The expanded form of \( \ln \sqrt{x} \) using properties of logarithms is \( 1/2 \ln x \) .

Step 1: Rewrite the square root as an exponent

Firstly, rewrite the square root of x as an exponent. The square root is equivalent to the 1/2 power, so the expression can be rewritten as \( \ln {x^{1/2}} \).

Step 2: Apply the power rule of logarithms

Now, apply the power rule to bring the exponent out in front. The rule states that for any logarithm, the exponent can be brought out as a coefficient. Therefore we get \( \ln {x^{1/2}} = 1/2 \ln x \).

Key Concepts

Properties of Logarithms

The properties of logarithms are like handy tools in mathematics, designed to make dealing with logarithmic expressions much simpler. They're based on arithmetic rules and can help convert complex logs into simpler or expanded forms.

Some key properties include:

• Product Rule: \( \log_b(M \times N) = \log_b(M) + \log_b(N) \)
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Rule: \( \log_b(M^n) = n \times \log_b(M) \)

These rules provide ways to manipulate and solve logarithmic expressions without getting too bogged down by complexity. In our exercise, the power rule is especially useful, allowing us to simplify and expand the expression easily.

Power Rule

The power rule is one of the properties of logarithms that significantly simplifies expressions. When you have a logarithm with an exponent, this rule allows you to "pull out" the exponent to the front of the logarithm.

For example, if you have \( \log_b(x^n) \), using the power rule, you can rewrite this as \( n \times \log_b(x) \). Basically, this turns a potentially cumbersome operation into a simple multiplication.

In our problem, we had \( \ln{x^{1/2}} \), so applying the power rule gives us \( \frac{1}{2} \ln{x} \). This makes the expression more accessible, allowing further evaluations or simplifications if necessary.

Square Root as Exponent

A square root might seem simple, but when you start manipulating expressions in algebra, it's often helpful to express roots as exponents.

The square root of a number \( x \) is equivalent to taking \( x \) to the power of \( \frac{1}{2} \). So, \( \sqrt{x} = x^{1/2} \).

This conversion is crucial because it enables us to apply logarithm properties - like the power rule - that depend on exponents being visible as such. In solving our exercise, recognizing that \( \sqrt[2]{x} = x^{1/2} \) allowed us to use the power rule efficiently, simplifying the original expression.

Natural Logarithm

The natural logarithm, denoted \( \ln \), is the logarithm with the base \( e \), where \( e \) is approximately equal to 2.71828. The choice of base \( e \) makes natural logs incredibly useful in calculus, exponential growth, and many natural processes.

Mathematically, \( \ln(x) = \log_e(x) \). Using natural logarithms, we can elegantly express growth phenomena and continuously compounding interest, among other things.

In our exercise, the expression \( \ln \sqrt{x} \) simplifies using the properties of natural logs just like any other logarithmic base, showcasing their versatility in handling various mathematical transformations.