Problem 23 - College Algebra and Calculus: An Applied Approach

Question

Write the logarithm in terms of natural logarithms.\(\log _{2.6} x\)

Step-by-Step Solution

Answer

The natural logarithm of \( \log _{2.6} x \) is \( \frac{{\ln x}}{{\ln 2.6}} \)

Step 1: Understanding the Change of Base Formula

The change of base formula is an identity used in logarithms asking for a change from one base to another. The formula is generally represented as \( \log_b a = \frac{{\ln a}}{{\ln b}} \), where 'a' is the argument and 'b' is the base of the logarithm.

Step 2: Apply the Change of Base Formula

Using the given equation, apply the change of base formula: \( \log _{2.6} x = \frac{{\ln x}}{{\ln 2.6}} \)

Key Concepts

Natural logarithms

The natural logarithm, often denoted as \( \ln \), is a special type of logarithm where the base is the mathematical constant \( e \), approximately equal to 2.718. Natural logarithms are frequently used in mathematics and science because of their unique properties and ease of use in calculus. In fact, they play a vital role in equations involving exponential growth or decay, as \( e \) is the base of the natural exponential function.

• Natural logarithms simplify calculations involving compound interest, population growth, and radioactive decay.
• One major feature is that the derivative of \( \ln(x) \) is \( \frac{1}{x} \), which offers simplicity in differentiation.
• The integral of \( \frac{1}{x} \) is \( \ln|x| + C \), adding another layer to their significance in calculus.

By using natural logarithms, complex problems in various fields become more manageable.

Logarithms

Logarithms are the inverse operation of exponentiation and are essential in fields such as engineering, physics, and computer science. The notation \( \log_b(a) \) refers to the logarithm of \( a \) with base \( b \), meaning "to what power must \( b \) be raised to yield \( a \)." Logarithms have several useful properties:

• They transform multiplication into addition, making complex multiplications easier to manage.
• They convert exponentiation into multiplication, which helps in solving exponential equations.
• They have a wide variety of applications, including sound intensity measurement in decibels and the Richter scale for earthquake strength.

Furthermore, logarithms extend to non-integer values with the help of real number bases, where ordinary algebraic methods would fall short. Using logarithms simplifies complicated expressions into comprehensible parts, which is an invaluable tool in mathematical problem-solving.

Base of logarithm

The base of a logarithm is the number that gets raised to a power in order to reach another number. In the notation \( \log_b(a) \), \( b \) is the base. Changing the base of a logarithm can be essential for simplifying expressions and making complex calculations more approachable. The change of base formula, \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \), allows for the transformation of logarithms of any given base into natural logarithms (base \( e \)).

• This formula simplifies comparing logarithms with different bases by converting them into a form that uses the natural logarithm.
• It becomes particularly helpful in computational environments where calculations involving different bases are needed.
• By using a common base, such as the natural logarithm base \( e \), we avoid compatibility issues in varied mathematical contexts.

This change of base technique is fundamental to accurately handling logarithmic expressions across diverse bases without losing the integrity of results.