Problem 67 - Precalculus with Calculus Previews

Question

In Problems \(57-68\), use \(\log _{b} 4=0.6021\) and \(\log _{b} 5=\) \(0.6990\) to evaluate the given logarithm. Round your answer to four decimal places. $$ \log _{4} b $$

Step-by-Step Solution

Answer

\( \log_{4} b \approx 1.6606 \).

Step 1: Understand the Problem

We need to evaluate the logarithm \( \log_{4} b \) using the given logarithmic values \( \log_{b} 4 = 0.6021 \) and \( \log_{b} 5 = 0.6990 \). We should express \( \log_{4} b \) in terms of logarithms with base \( b \).

Step 2: Use Change of Base Formula

The change of base formula for logarithms is \( \log_a c = \frac{\log_b c}{\log_b a} \). We apply this formula to \( \log_{4} b \):\[\log_{4} b = \frac{\log_{b} b}{\log_{b} 4}\]

Step 3: Substitute Known Values

We know that \( \log_b b = 1 \) because any logarithm of a number to the same base is 1. Substitute the given value for \( \log_{b} 4 \):\[\log_{4} b = \frac{1}{0.6021}\]

Step 4: Perform the Calculation

Compute the value:\[\log_{4} b \approx \frac{1}{0.6021} \approx 1.6606\]

Step 5: Round Off

Since we need the answer rounded to four decimal places, our calculated value \( 1.6606 \) is already in that form.

Key Concepts

Change of Base Formula

The change of base formula is a significant tool in algebra and allows us to evaluate logarithms of any base using logarithms of a base we are more familiar or comfortable with. The common form of this formula is:

• \( \log_a c = \frac{\log_b c}{\log_b a} \)

This formula becomes particularly useful when the logarithm base is not a common one like 10 or \( e \), for which calculators often have dedicated buttons. For instance, in our problem, converting \( \log_{4} b \) into a logarithm with base \( b \) helps use the known values \( \log_{b} 4 = 0.6021 \) and \( \log_{b} 5 = 0.6990 \) to find the solution.
To apply it accurately:

• Identify the base of the logarithm in the expression you wish to convert.
• Find the logarithm to be evaluated in terms of a new base using the formula.
• Substitute any known values to simplify and solve the expression.

Logarithmic Identities

Logarithmic identities are sets of rules that allow for the manipulation and simplification of logarithms. One of the key identities used in our problem is \( \log_b b = 1 \), which states that the logarithm of a number to its own base is equal to one. This identity simplifies calculations whenever we come across a logarithm where the base and the number are the same.
Moreover, here are a few other essential logarithmic identities:

• \( \log_b (xy) = \log_b x + \log_b y \)
• \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• \( \log_b (x^n) = n \cdot \log_b x \)

Identities like these help transform complex expressions into simpler ones, making them easier to evaluate or rearrange, just as in our solution process for \( \log_{4} b \). Always keep these handy as tools to break down more challenging expressions!

Base of Logarithm

The base of a logarithm is a fundamental concept in understanding and working with logarithmic expressions. Simply put, it is the number that is raised to a power to produce a given number. For example, in \( \log_4 b \), the base is 4, and we want to find what exponent we need for 4 to yield \( b \).
Some bases are more common than others:

• Base 10: Often called the 'common logarithm', used frequently in scientific calculations (\( \log_{10} \)).
• Base \( e \): Known as the 'natural logarithm', and is used extensively in calculus and mathematical modeling (\( \ln \) or \( \log_e \)).
• Specific number bases: As in our exercise with base 4, these are less common but can still be evaluated using the change of base formula.

Understanding the base is crucial as it determines the scale or magnitude alterations in values represented logarithmically. Always consider it as the root or foundation when evaluating or rewriting such expressions.