Problem 65 - Precalculus Mathematics for Calculus

Question

Use the Change of Base Formula to show that $$\log e=\frac{1}{\ln 10}$$

Step-by-Step Solution

Answer

Using the change of base formula, \(\log e = \frac{\ln e}{\ln 10} = \frac{1}{\ln 10}\).

Step 1: Recall the Change of Base Formula

The change of base formula for logarithms allows us to convert a logarithm of any base to a different base using the formula: \[ \log_b a = \frac{\log_k a}{\log_k b}\] where \(b\) is the original base and \(k\) is the new base. We will use base \(k = 10\) in our problem.

Step 2: Apply the Change of Base Formula

In our problem, we are given \(\log e\) with base 10 implicitly understood, so we apply the change of base formula: \[\log e = \frac{\ln e}{\ln 10}\] Where \(\ln e\) is the natural log of \(e\).

Step 3: Evaluate \(\ln e\)

We know that \(\ln e = 1\) because the natural logarithm is the logarithm to the base \(e\), and the logarithm of a number with its base is 1. So, replace \(\ln e\) with 1 in the equation:\[\log e = \frac{1}{\ln 10}\]

Step 4: Conclude

Thus, using the properties of logarithms and the change of base formula, we find that:\[\log e = \frac{1}{\ln 10}\] This is consistent with the result we set out to prove.

Key Concepts

Logarithms

Logarithms are powerful mathematical tools that help us deal with exponential relationships. They are essentially the opposite of exponents. When we say \( ext{log}_b(a)\), we are asking the question: to what power must the base \(b\) be raised to yield the number \(a\)? For example, \( ext{log}_2(8) = 3\) because \(2^3 = 8\).
Understanding logarithms requires knowing a few key properties:

• Product Rule: \( ext{log}_b(xy) = ext{log}_b(x) + ext{log}_b(y)\)
• Quotient Rule: \( ext{log}_b(x/y) = ext{log}_b(x) - ext{log}_b(y)\)
• Power Rule: \( ext{log}_b(x^n) = n\text{log}_b(x)\)

The change of base formula is another crucial property, which allows for the conversion of a logarithm from one base to another, specifically using a base that might be easier or more convenient to work with, such as base 10 or base \(e\).

Natural Logarithm

The natural logarithm, denoted \( ext{ln}\), is a logarithm where the base is the mathematical constant \(e\) (approximately 2.71828). In mathematical terms, \( ext{ln}(x) = ext{log}_e(x)\).
One of the unique uses of natural logarithms is their occurrence in situations involving continuous growth or decay, such as in population models or radioactive decay.

Some properties of natural logarithms include:

• extbf{ln}(e) = 1: Because the logarithm of a number to its own base is always 1.
• extbf{ln}(1) = 0: Since any number raised to the power of 0 is 1.

The natural logarithm is particularly useful because it simplifies a lot of calculus operations and naturally appears in several mathematical contexts due to the unique properties of \(e\).

Base Conversion

Base conversion in logarithms is essential when dealing with different systems and calculations. The change of base formula for logarithms facilitates this conversion, allowing us to calculate logarithms even when the base isn't compatible with our given operations.
For our problem, understanding base conversion required the transformation of \( ext{log}(e)\) using a new base, which involves the change of base formula:

• Formula: \( ext{log}_b(a) = \frac{\text{log}_k(a)}{\text{log}_k(b)}\)

In this particular exercise, we convert the logarithm of \(e\) with an implicit base of 10 to the natural log \( ext{ln}\) base. This is achieved by realizing that \(\text{ln}(e)\) equals 1, leading to the simplification we sought: \( ext{log}(e) = \frac{1}{\text{ln}(10)}\). This concept is vital for efficiently tackling problems involving logarithms with unfamiliar bases.