Problem 57 - Algebra A Combined Function

Question

If \(\log _{b} 3=0.5\) and \(\log _{b} 5=0.7,\) evaluate each expression. $$ \log _{b} \sqrt{5} $$

Step-by-Step Solution

Answer

\( \log_b \sqrt{5} = 0.35 \).

Step 1: Recall the Square Root Property of Logarithms

The square root property of logarithms states that \( \log_b \sqrt{x} = \frac{1}{2} \log_b x \). This is derived from the power rule \( \log_b (x^n) = n \log_b x \), where \( n = \frac{1}{2} \).

Step 2: Apply the Square Root Property

We need to evaluate \( \log_b \sqrt{5} \). By the square root property, this is \( \log_b \sqrt{5} = \frac{1}{2} \log_b 5 \).

Step 3: Substitute the Given Logarithm Value

We know from the problem statement that \( \log_b 5 = 0.7 \). Substitute this value into the expression from the previous step: \( \log_b \sqrt{5} = \frac{1}{2} \times 0.7 \).

Step 4: Calculate the Final Expression

Perform the multiplication: \( \frac{1}{2} \times 0.7 = 0.35 \). Therefore, \( \log_b \sqrt{5} = 0.35 \).

Key Concepts

Square Root Property of Logarithms

In the realm of logarithms, the square root property is an essential tool. It provides a shortcut to simplify logarithms involving square roots. Specifically, the square root property states that for any base \( b \) and a positive number \( x \), the logarithm of the square root of \( x \) is half the logarithm of \( x \) itself.

In mathematical terms, it is expressed as:

• \( \log_b \sqrt{x} = \frac{1}{2} \log_b x \)

This property leverages the principle behind exponents, as square roots can be written as fractional exponents (i.e., \( x^{1/2} \)). By applying the power rule of logarithms, which will be discussed shortly, we are able to transform the expression of the square root into a form that's easier to handle.

Understanding the square root property allows us to process more complex logarithmic expressions efficiently, especially when dealing with roots. It provides a powerful way to break down complicated equations into solvable terms.

Power Rule in Logarithms

The power rule is a fundamental concept when working with logarithmic expressions. It states that the logarithm of a number raised to an exponent can be simplified by multiplying the exponent by the logarithm of the base number. This can be formulated as:

• \( \log_b (x^n) = n \cdot \log_b x \)

This rule is incredibly useful when dealing with exponential functions. By applying the power rule, you can take the exponent out of the logarithm, making it easier to work with.

In our exercise, the power rule helps to transition from \( \log_b \sqrt{5} \) to \( \frac{1}{2} \log_b 5 \) by understanding the square root as an exponent: \( 5^{1/2} \). By using the power rule, the problem simplifies to a multiplication of the constant \( \frac{1}{2} \) with the already known logarithm \( \log_b 5 = 0.7 \).

The power rule is one of the cornerstones of logarithmic manipulation, enabling you to reframe expressions in a more manageable form.

Understanding Logarithmic Expressions

Logarithmic expressions are equations that contain logarithms, which are the inverse operations of exponentiation. The core idea is to find out the exponent that yields a certain number when used as the power of a given base. In simpler terms, for \( \log_b x = y \), \( b \) raised to the power \( y \) equals \( x \) ( \( b^y = x \) ).

Engaging with logarithmic expressions involves utilizing properties like the square root property and power rule. These properties convert complex forms of logarithms into simpler, and often more intuitive, pieces.

In the exercise provided, knowing the values of \( \log_b 3 = 0.5 \) and \( \log_b 5 = 0.7 \) provides the necessary components for evaluating other logarithmic expressions, such as \( \log_b \sqrt{5} \). With these given values and the properties discussed, you can easily substitute and compute the desired outcomes.

Understanding how to manipulate and evaluate these expressions is key in algebra and calculus, as it lays the groundwork for solving equations and modeling real-world phenomena efficiently.