Problem 15 - Algebra 2

Question

Evaluate each logarithm. $$ \log _{4} 2 $$

Step-by-Step Solution

Answer

The value of \(\log _{4} 2\) is 1/2.

Step 1: Understanding Logarithm Properties

A property of logarithms states that \(b^{\log_b{a}} = a\), where \(b\) is the base and \(a\) is the argument. We will use this property to solve the problem.

Step 2: Applying the Property to Solve the Logarithm

If we set \(4^x = 2\), we can evaluate \(x\) by recalling that \(4 = 2^2\). Hence, \(2^2 \)^x = 2. The bases are the same, so the exponents must be equal, which gives us that \(x = 1/2\).

Step 3: Confirming the Answer

We could also verify this by substitution. If we substitute 1/2 into \(4^x = 2\), we get \(4^{1/2} = 2\), which holds true because the square root of 4 is indeed 2. Therefore, \(\log_4 {2} = 1/2 \).

Key Concepts

Exponents

Exponents are a fundamental concept essential for understanding logarithms. They indicate how many times a number, known as the base, is multiplied by itself. For example, in the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent, telling us to multiply \(a\) by itself \(n\) times.
The inverse operation of exponents is key to logarithms. For instance, the exponent in \(4^x = 2\) is what we're solving for in the logarithm \(\log_4 2\). This means finding the power we have to raise 4 to get 2. In solving logarithmic equations, an understanding of exponents as repeated multiplication helps relate numbers, particularly when matching and manipulating the bases.

Base of a Logarithm

The base of a logarithm is a crucial part of a log expression. It determines what number is repeatedly multiplied to result in the given argument or number. In \(\log_b a\), \(b\) is the base. The exercise \(\log_4 2\) implies that 4 is the base.
This base shows that we are finding out "how many times we multiply" the base number (4) to get to 2. Understanding the base allows us to make transformations, like recognizing that \(4 = 2^2\), which is beneficial for simplifying and solving equations. The base heavily influences the methods we apply, such as using properties of exponents to solve logarithmic equations.

Logarithmic Properties

Logarithmic properties simplify the manipulation and solving of logarithmic expressions. One key property is the change of base, which tells us that \(b^{\log_b a} = a\). This means the base \(b\) raised to the power of its logarithm results in the number \(a\) itself.
In our specific problem, the property is applied as we set \(4^x = 2\). By recognizing that \(4\) can be expressed as \(2^2\), we can solve for \(x\) by equating exponents, resulting in \(x = \frac{1}{2}\). Additionally, learning to manipulate logarithms involves understanding properties like the power rule, product rule, and others which provide shortcuts and insights into simplifying complex equations. Knowing these properties enables more intuitive and quicker solutions to seemingly complicated logarithmic problems.