Problem 3 - Precalculus with Limits

Question

In Exercises \(1-3,\) fill in the blanks. You can consider log \(_{a} x\) to be a constant multiple of \(\log _{b} x ;\) the constant multiplier is _____ .

Step-by-Step Solution

Answer

The constant multiplier is \(\frac{1}{\log_{b} a}\).

Step 1: Understand the Problem

In a logarithm, the base can be changed using a specific formula known as the change of base formula, which is \(\log_b a = \frac{1}{\log_a b}\). Here, we are seeking the multiplier that enables us to change the base from \(b\) to \(a\).

Step 2: Substitute using the Change Base Formula

Looking at the formula given in the question, it can be seen as an application of the change of base formula. We can then rewrite \(\log_{a}x\) as \(\frac{1}{\log_{b} a} \log_{b} x\). Upon rearranging, it takes the form of \(\log_{a}x = M \log_{b}x\), where \(M = \frac{1}{\log_{b} a}\) is the multiplier.

Step 3: Identify the Multiplier

After the above step, it becomes clear that the multiplier \(M\) for the expression is \(\frac{1}{\log_{b} a}\).

Key Concepts

Logarithms

Logarithms are mathematical functions that help us solve equations involving exponents. When you think of a logarithm, consider it as the opposite of an exponent. If exponentiation is about multiplying a number by itself a certain number of times, then a logarithm tells us how many times a base number must be multiplied by itself to reach a given number.

Logarithms are expressed as \( \log_b x \), which reads as "log base \( b \) of \( x \)." Here, \( b \) is the base of the logarithm, and \( x \) is the number you want to find the logarithm of. For example, \( \log_2 8 \) is asking, "to what power should we raise 2 to get 8?" The answer is 3, because \( 2^3 = 8 \).

Key points to remember about logarithms include:

• They help solve exponentiation problems by reversing the operation.
• The base usually needs to be a positive number different from 1.
• Common bases include 10 and e, but you can use any positive number as a base.

Logarithmic Properties

Logarithmic properties simplify complex expressions and solve logarithmic equations more efficiently. These properties derive from the rules of exponents and are essential for manipulating logarithmic expressions.

Some vital properties of logarithms include:

• Product Rule: This states that \( \log_b (xy) = \log_b x + \log_b y \). It tells us the logarithm of a product is the sum of the logarithms.
• Quotient Rule: This property is \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \). It states that the logarithm of a quotient is the difference of the logarithms.
• Power Rule: According to this rule, \( \log_b (x^n) = n \cdot \log_b x \). This helps us deal with exponents inside a logarithm.
• Change of Base Rule: The formula \( \log_b a = \frac{1}{\log_a b} \) is particularly useful. It allows us to change the base of a logarithm, which is helpful in many mathematical computations.

Understanding these properties gives you powerful tools to tackle logarithmic problems with confidence.

Mathematical Constants

Mathematical constants are specific numbers with significant importance in various fields of mathematics, including calculus and logarithms. They have a precise and unchanging value.

Some of the constants frequently encountered with logarithms are:

• Base 10 Constant (Common Logarithms): This is often written as \( \log_{10} x \). Common logarithms are used widely in science and engineering.
• Base e Constant (Natural Logarithms): Expressed as \( \ln x \), which is the natural logarithm where \( e \approx 2.71828 \). The constant \( e \) is fundamental in calculus due to its unique properties in growth and decay models.

These constants play crucial roles in simplifying problems and in theoretical discussions within mathematics and beyond. They help create universal standards across disciplines, making complex calculations more accessible.