Problem 42 - Precalculus

Question

Evaluate the logarithms exactly (if possible). $$\log _{1 / 7} 2401$$

Step-by-Step Solution

Answer

The exact value of \( \log _{1/7} 2401 \) is -4.

Step 1: Understanding the Problem Statement

We need to evaluate the logarithm \( \log _{1/7} 2401 \), and find the exact value if possible. This means we need to determine the exponent to which the base \( \frac{1}{7} \) must be raised to obtain 2401.

Step 2: Express 2401 as a Power of 7

First, recognize that 2401 is a power of 7. We calculate powers of 7 until we get 2401: \( 7^4 = 2401 \). Thus, \( 2401 = 7^4 \).

Step 3: Change of Base and Simplification

The logarithmic expression \( \log _{1/7} 2401 \) can be rewritten using the identity for changing the base: \(\log _{1/7} 2401 = \frac{\log _7 2401}{\log _7 \left( \frac{1}{7} \right)}\). We already know \(\log _7 2401 = 4\) since \(2401 = 7^4\).

Step 4: Evaluate the Logarithm of the Reciprocal Base

Determine \(\log _7 \left( \frac{1}{7} \right)\). The expression \( \frac{1}{7} = 7^{-1} \), so \(\log _7 (7^{-1}) = -1\).

Step 5: Final Calculation

Substitute the known values into the formula: \( \frac{\log _7 2401}{\log _7 \left( \frac{1}{7} \right)} = \frac{4}{-1} = -4 \). The exact value of \( \log _{1/7} 2401 \) is -4.

Key Concepts

Change of Base Formula

The change of base formula is a fundamental tool in evaluating logarithms of any base, especially when you wish to express it in a base more suited to your needs or a calculator's capabilities. It allows us to convert a logarithm with one base to an equivalent expression with a different base.
The formula is expressed as follows:

• \[\log_a b = \frac{\log_c b}{\log_c a}\]

This means that the logarithm of a number \( b \) with base \( a \) can be rewritten as a division between the logarithm of \( b \) and the logarithm of \( a \), both using a new base \( c \). Commonly, the new base \( c \) is chosen to be either 10 or \( e \) (natural logarithm), because these can easily be calculated with most scientific calculators.
In our example of \( \log_{1/7}2401 \), we used the change of base formula to rewrite it as:

• \[\log_{1/7} 2401 = \frac{\log_7 2401}{\log_7 (1/7)}\]

This allows us to comfortably handle it in terms of base 7, which we already explored.

Exponents

Understanding exponents is essential when working with logarithms because logarithms are the inverses of exponential functions. An exponent refers to the number that indicates how many times a base number is multiplied by itself. For example, in the expression \( 7^4 \), 7 is the base, and 4 is the exponent, indicating that 7 is multiplied 4 times: \( 7 \times 7 \times 7 \times 7 = 2401 \).
When solving \( \log_{1/7}2401 \), recognizing that 2401 is equivalently expressed as \( 7^4 \) is key. It allows us to link the logarithm with known exponential values. This connection is crucial because the property of logarithms is that they reflect what exponent a base is raised to achieve a certain number.
In other cases, you might need to break down a number into its base power form, as we did with 2401, to evaluate the logarithm directly.

Properties of Logarithms

The properties of logarithms make calculations easier and more manageable, especially when dealing with complex expressions. One of the fundamental properties is the power rule: \( \log_b(a^c) = c \cdot \log_b a \). This indicates that if a number is expressed in exponential form, the exponent can be moved forward as a coefficient.
For example, in our problem, we know:

• \[\log_7 (7^4) = 4 \times \log_7 7 = 4\]

This is simplified further due to another property that any number's logarithm to its own base is 1: \( \log_b b = 1 \). Thus, \( \log_7 7 = 1 \), and it elegantly provides the result.
Another key property is handling the logarithm of a reciprocal base, such as \( \log_7 (1/7) \). Knowing that \( 1/7 \) can be written as \( 7^{-1} \), we use the power rule to find:

• \[\log_7 (7^{-1}) = -1 \times \log_7 7 = -1\]

These properties are powerful tools in simplifying logarithmic expressions and understanding how they relate to their exponential roots.