Problem 25 - Intermediate Algebra

Question

Show that \(\log _{b^{2}} x=\frac{1}{2} \log _{b} x\)

Step-by-Step Solution

Answer

The equality holds true as \(\log_{b^2} x = \frac{1}{2}\log_b x\) simplifies to the same expression \(\frac{\log_b x}{2}\).

Step 1: Understand the Logarithmic Identities

Recall that the logarithm properties include \(\log_a b^m = m \log_a b\). This means that a power inside a logarithm can be factored out front as a multiplication. These properties will help simplify the expression.

Step 2: Apply the Change of Base Formula

Use the change of base formula which states \(\log_{b^2} x = \frac{\log_b x}{\log_b b^2}\). We can simplify the denominator further by using logarithm properties.

Step 3: Simplify the Denominator

Applying \(\log_b b^2 = 2 \log_b b = 2\) because \(\log_b b = 1\). Thus, the denominator simplifies to \(2\). Now, the expression becomes \(\log_{b^2} x = \frac{\log_b x}{2}\).

Step 4: Compare with the Right Side

Rewrite the right side of the initial equation: \(\frac{1}{2} \log_b x\). Notice that this expression is identical to \(\frac{\log_b x}{2}\).

Step 5: Conclude the Equality

Both expressions, \(\log_{b^2} x\) and \(\frac{1}{2} \log_b x\), simplify to \(\frac{\log_b x}{2}\). Therefore, \(\log_{b^2} x = \frac{1}{2}\log_b x\) is true as both sides are equal.

Key Concepts

Logarithmic Identities

Logarithmic identities are essential tools in simplifying and manipulating logarithmic expressions. One crucial identity is the power rule: \(\log_a b^m = m \cdot \log_a b\). This identity tells us that when you have a power inside the logarithm, you can bring the exponent down in front of the logarithm as a multiplier.

This property is highly useful in various calculations and helps in breaking down complex expressions, making them easier to work with. For example, when dealing with an expression like \(\log_b b^2\), the power rule allows us to simplify it to \(2\cdot \log_b b\), which further simplifies to \(2\cdot 1 = 2\) because \(\log_b b = 1\). This step is particularly useful in transforming the base of logarithmic expressions when they appear with squared bases or exponents.

Change of Base Formula

The change of base formula is a handy tool when transforming between different logarithmic bases. Often, you might need to evaluate logarithms with bases that are not immediately convenient or available on calculators. The formula states:

• \(\log_c a = \frac{\log_b a}{\log_b c}\)

This means you can convert a logarithm with base \(c\) to a new base \(b\), allowing more flexibility in calculations.

As seen in our exercise, converting \(\log_{b^2} x\) using the change of base formula becomes:

• \(\log_{b^2} x = \frac{\log_b x}{\log_b b^2}\)

By applying the logarithmic identity \(\log_b b^2 = 2\), the expression simplifies to \(\frac{\log_b x}{2}\). This demonstrates how the change of base formula simplifies expressions and helps in solving equations involving unusual bases.

Simplifying Expressions

Simplifying logarithmic expressions is key to solving equations efficiently. Understanding the properties and identities of logarithms is crucial in this process. In the exercise, we use simplification techniques to resolve \(\log_{b^2} x\) to \(\frac{1}{2} \cdot \log_b x\).

• By employing the change of base and the power rule, complex expressions become manageable.
• Consider the expression \(\frac{\log_b x}{\log_b b^2}\): simplifying \(\log_b b^2\) to \(2\), reduces the expression to \(\frac{\log_b x}{2}\).

When comparing this to \(\frac{1}{2} \cdot \log_b x\), it becomes clear the expressions are identical, confirming that the equation holds true.

The skill of simplifying expressions efficiently allows you to solve logarithmic equations more easily and accurately, enhancing your problem-solving capabilities.