Problem 27 - Calculus and its applications

Question

Given \(\log _{b} 3=1.099\) and \(\log _{b} 5=1.609\), find each value. $$ \log _{b} \sqrt{b^{3}} $$

Step-by-Step Solution

Answer

\( \log_{b} \sqrt{b^3} = \frac{3}{2} \).

Step 1: Understand the Expression

The expression given is \( \log_{b} \sqrt{b^3} \). This means we are looking for the logarithm of \( \sqrt{b^3} \) to the base \(b\).

Step 2: Simplify the Square Root

Recognize that \( \sqrt{b^3} \) can be rewritten in its exponential form. The square root of \( b^3 \) is \( (b^3)^{1/2} \), which simplifies to \( b^{3/2} \).

Step 3: Apply Logarithm Power Rule

Use the logarithm power rule, which states \( \log_{b} (b^n) = n \cdot \log_{b} (b) = n \). Thus, \( \log_{b} b^{3/2} = \frac{3}{2} \).

Step 4: Combine the Steps

From Step 3, we conclude that \( \log_{b} \sqrt{b^3} = \frac{3}{2} \), because any logarithm of a base to its own power equals the power itself.

Key Concepts

Logarithm Properties

Logarithms are an important concept in mathematics. They simplify complex calculations by transforming multiplicative relationships into additive ones. This is because of the logarithm property known as the product rule.

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient Rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
• Change of Base Formula: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \), allows us to convert logs to different bases.

Another property to note is the base property. This property is intuitive and useful: \( \log_b(b) = 1 \). When dealing with the logarithms of numbers that are powers of the base, this property helps to pinpoint the exact value quickly.

Exponential Form

Exponential form is a powerful way to represent numbers, making it easier to work with powers and roots. In this form, a number like \( b^n \) indicates that the base \( b \) is multiplied by itself \( n \) times.
Let's consider the problem where we need to find the logarithm of \( \sqrt{b^3} \). Here, the square root can be rewritten using exponential form as \((b^3)^{1/2} = b^{3/2}\).
This transformation is helpful because it allows us to use logarithm rules more easily. Expressing complex operations as powers can simplify calculations and deepen understanding of exponential growth or decay.

Logarithm Power Rule

The logarithm power rule is an essential tool for simplifying logs that have a number raised to a power:
\[ \log_b (x^n) = n \cdot \log_b(x) \]Using this rule can turn multiplications inside the log into multiplications outside, simplifying the problem-solving process. In our exercise, we have \( \log_b(b^{3/2}) \).

• Application: When we apply the power rule, it tells us that this is simply \(\frac{3}{2} \cdot \log_b(b) \). Knowing that \( \log_b(b) = 1 \), we quickly resolve this to \(\frac{3}{2}\).
• This approach shows that even complex-looking expressions can break down into simpler parts. This helps save time and builds a clearer picture of the whole log structure.

Understanding the logarithm power rule empowers you to deal with a wide range of logarithmic functions efficiently and accurately.