Problem 25 - Calculus and its applications

Question

Given \(\log _{b} 3=1.099\) and \(\log _{b} 5=1.609\), find each value. $$ \log _{b} \frac{1}{5} $$

Step-by-Step Solution

Answer

\( \log _{b} \frac{1}{5} = -1.609 \)

Step 1: Use the Properties of Logarithms

Recall that the logarithm of a reciprocal can be expressed as the negative of the logarithm. Therefore, \[ \log _{b} \frac{1}{5} = -\log _{b} 5. \] This is based on the logarithmic identity: \( \log_{b} \left( \frac{1}{x} \right) = - \log_{b} x \). Using this, we can simplify the expression.

Step 2: Substitute Known Values

Substitute the given value of \( \log_{b} 5 \) into the equation:\[ \log _{b} \frac{1}{5} = -1.609. \] Since \( \log _{b} 5 = 1.609 \), replace \( \log _{b} 5 \) with 1.609 in the equation, leaving us with the final calculation for the logarithm of the fraction.

Key Concepts

Understanding the Properties of Logarithms

Logarithms have several fundamental properties that simplify complex expressions and calculations. By leveraging these properties, we can break down tough problems into manageable steps. Here are a few important properties to remember:

• **Product Property:** \( \log_{b}(xy) = \log_{b} x + \log_{b} y \). This means you can add two logarithms together if their arguments are multiplied.
• **Quotient Property:** \( \log_{b}(\frac{x}{y}) = \log_{b} x - \log_{b} y \). Use this property when the arguments are divided.
• **Power Property:** \( \log_{b}(x^n) = n \log_{b} x \). Raise a number to a power and you've multiplied the logarithm by that power.
• **Reciprocal Property:** \( \log_{b} \left( \frac{1}{x} \right) = -\log_{b} x \). This tells us that inverses change the sign of the logarithm.

These properties are incredibly useful for breaking down expressions and solving logarithmic equations. They allow us to manipulate and simplify expression forms, making problem-solving more intuitive.

Performing Logarithmic Calculations

When trying to determine the logarithm of a fractional expression like \(\log _{b} \frac{1}{5}\), apply the reciprocal property. This property states you should negate the logarithm of the denominator:
\[ \log _{b} \frac{1}{5} = -\log _{b} 5.\]
Using the problem's given value \( \log_{b} 5 = 1.609 \), substitute and calculate: \[ \log _{b} \frac{1}{5} = -1.609.\]
General steps for logarithmic calculations involve:

• Identifying which properties apply to the expression.
• Simplifying the expression using those properties.
• Substituting known values to find the result.

The process highlights how understanding properties aids in efficient calculations, and shifts complex logarithmic problems into straightforward tasks.

Exploring the Base of Logarithms

In logarithmic expressions, the base defines what number you raise to power to reach another. In this exercise, the base is referred to as \(b\). The base influences how we think about exponentiation and logarithms:

• **Common Logarithm:** Uses base 10, expressed simply as \(\log\). It's great for metric calculations and quick approximations.
• **Natural Logarithm:** Uses base \(e\) (approximately 2.718), expressed as \(\ln\). A staple in calculus, it bridges continuous growth and complex analysis.
• **Custom Base Logs:** Allow more versatile problem-solving, adapting to different contexts as in this problem where the base \(b\) is not explicitly defined, but essential for accurate calculation.

Knowing the base helps to properly apply logarithmic properties and ensure correct substitution of known values during calculations. Each base has specific scenarios where it excels, aiding understanding of diverse mathematical problems.