Problem 81 - Precalculus with Calculus Previews

Question

If \(a>0\) and \(b>0, a \neq b\), then \(\log _{a} x\) is a constant multiple of \(\log _{b} x .\) That is, \(\log _{a} x=k \log _{b} x .\) Find \(k\).

Step-by-Step Solution

Answer

The constant \( k \) is \( \log_b a \).

Step 1: Definition of Logarithms

Recall the definition of a logarithm: if \( y = \log_a x \), it means that \( a^y = x \). Similarly, if \( z = \log_b x \), it means that \( b^z = x \).

Step 2: Change of Base Formula

To compare \( \log_a x \) with \( \log_b x \), use the change of base formula: \( \log_a x = \frac{\log_c x}{\log_c a} \) and \( \log_b x = \frac{\log_c x}{\log_c b} \) for any positive base \( c \), commonly using \( c = 10 \) or \( c = e \).

Step 3: Express \( \log_a x \) Using \( \log_b x \)

Apply the change of base formula: \( \log_a x = \frac{\log_c x}{\log_c a} \) and \( \log_b x = \frac{\log_c x}{\log_c b} \). This means \( \log_a x = \log_b x \cdot \frac{\log_c b}{\log_c a} \) after equating the inner parts.

Step 4: Identify the Constant Multiple \(k\)

The expressions give us \( \log_a x = k \log_b x \), where \( k = \frac{\log_c b}{\log_c a} \). By properties of logarithms, this simplifies to \( k = \log_b a \).

Key Concepts

Change of Base Formula

The change of base formula is a useful tool for evaluating logarithms with bases that may not be immediately compatible with common computational systems, like calculators that typically use base 10 or base \(e\).
To use the change of base formula, you express a logarithm with an unfamiliar base as a ratio of logarithms with a more familiar base:

• For example, \( \log_a x \) can be written as \( \frac{\log_c x}{\log_c a} \).
• The formula allows you to use any positive base \( c \), however, base 10 and base \(e\) (natural logarithm) are frequently used in practice.

This transformation is essential in situations where the initial base does not easily lend itself to calculation, enabling more flexible and versatile problem-solving approaches.

Properties of Logarithms

Logarithms come with a set of properties that make them particularly valuable in solving mathematical equations involving exponential growth or decay and simplifying expressions.
Some key properties include:

• Product Property: \(\log_b (xy) = \log_b x + \log_b y\) shows how a logarithm of a product is the sum of logarithms.
• Quotient Property: \(\log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y\) illustrates the subtraction of logarithms for a division.
• Power Property: \(\log_b (x^p) = p \cdot \log_b x\) allows exponents within the logarithm argument to be moved in front as multiplication.
• Change of Base: Expresses a logarithm as a ratio of different bases, demonstrated previously in the change of base formula.

Understanding these properties aids in the manipulation and simplification of logarithmic equations, making them more manageable.

Constant Multiplication in Logarithms

Constant multiplication in logarithms arises when you express one logarithm as a multiple of another with a different base, as shown in the given problem.
In the original exercise, we determined that \(\log_a x = k \log_b x\) where \(k\) represents a constant. Through the use of the change of base formula, this constant \(k\) is found to be \(\log_b a\).
To find \(k\), you transform the bases of the logarithms using the formula:

• \(\log_a x = \frac{\log_c x}{\log_c a}\) and \(\log_b x = \frac{\log_c x}{\log_c b}\).
• The ratio \(\frac{\log_c b}{\log_c a}\) gives you \(k\), which simplifies to \(\log_b a\), revealing the constant multiple between the logarithms.

This concept often helps in making complex logarithmic comparisons more accessible and solving equations where different logarithmic bases are involved.