Problem 23 - Algebra 2

Question

Evaluate each logarithm. $$ \log _{\frac{1}{2}} \frac{1}{2} $$

Step-by-Step Solution

Answer

The value of \(\log _{\frac{1}{2}} \frac{1}{2}\) is 1.

Step 1: Understand the logarithm

First, it's essential to understand what \(\log _{\frac{1}{2}} \frac{1}{2}\) is asking for. This logarithm asks the question 'To what power must we raise \(\frac{1}{2}\) to obtain \(\frac{1}{2}\)?'.

Step 2: Apply the Identity Property of Logarithms

The identity property of logarithms states that the \(\log_b (b) = 1\). In this particular case \(b=\frac{1}{2}\). So applying identity property, \(\log _{\frac{1}{2}} \frac{1}{2} = 1\).

Step 3: Final step

So, \(\log _{\frac{1}{2}} \frac{1}{2}\) equals 1, which can be interpreted as '\(\frac{1}{2}\) raised to the power of 1 equals \(\frac{1}{2}\)'.

Key Concepts

Identity Property of Logarithms

The identity property of logarithms is a fundamental concept that simplifies working with logarithmic expressions. Essentially, this property tells us that if the base of the logarithm and the argument of the logarithm are the same, the result will always be 1.

• Mathematically, it is expressed as \( \log_b(b) = 1 \).
• This property answers the question of what exponent is needed to raise the base \(b\) to itself.
• In the example \( \log_{\frac{1}{2}} \frac{1}{2} \), applying this property gives us 1 because \( \frac{1}{2}^1 = \frac{1}{2} \).

Understanding this property allows us to quickly evaluate logarithmic expressions when the base and the argument are identical.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to the power of an exponent. It expresses repeated multiplication of the base.

• If you have \( b^x \), \(b\) is the base, and \(x\) is the exponent.
• For example, \( 2^3 = 2 \times 2 \times 2 = 8 \).
• Exponentiation is integral to understanding logarithms, as a logarithm is the inverse operation of exponentiation.

In the exercise, when you see \( \frac{1}{2}^1 = \frac{1}{2} \), it is showing that raising the base \(\frac{1}{2}\) to the power of 1 results in the original number, which connects back to the identity property of logarithms.

Base of a Logarithm

The base of a logarithm is a critical component in understanding how logarithms work.

• The base is the number that is raised to the power of the result of the logarithm.
• In the logarithmic form \( \log_b(y) \), \(b\) is the base.
• The base tells us the number we start with before applying the exponent to achieve the argument \(y\).

Having a clear understanding of what a base is in a logarithm helps when solving problems, especially when utilizing properties like the identity property. For example, in \( \log_{\frac{1}{2}} \frac{1}{2} \), the base is clearly \(\frac{1}{2}\), and it sets the context for identifying that the exponent must be 1 to satisfy the equation.