Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Write the logarithm in terms of natural logarithms.\(\log _{10} 20\)

Step-by-Step Solution

Answer

The common logarithm \(\log _{10} 20\) in terms of natural logarithms is \(\ln 20\)

Step 1: Identify the given values and the formula needed

In the problem, the values given are a=20 and b=10. We need the formula for a change of base, which is \(\log_b a=\frac{\log_k a}{\log_k b}\). We want to change the base to e, which is the base of natural logarithms, so k=e.

Step 2: Substitute the values into the formula

Substitute the given values into the formula, which will be \(\log_{10} 20=\frac{\ln 20}{\ln 10}\). Here, ln represents the natural logarithm, which is the logarithm to the base e.

Step 3: Simplify the expression

We know that \(\ln 10\) is equal to 1, as any log to the same base equals 1. Substitute this into the formula to get \(\log_{10} 20=\ln 20\).

Key Concepts

Natural Logarithms

Natural logarithms are a special type of logarithm where the base is the mathematical constant \(e\), approximately equal to 2.71828. The natural logarithm is often represented as \(\ln\). It is especially useful in calculus and mathematical modeling.

• Natural logarithms appear frequently in scientific calculations.
• They help simplify complex expressions, particularly those involving exponential growth or decay.
• \(e\) is an irrational number, meaning it cannot be expressed exactly as a fraction, and it is fundamental in mathematics.

Natural logarithms give us a way to better understand growth patterns. For example, calculating continuously compounded interest or modeling population growth over time are instances when \(\ln\) can be very helpful. It plays a crucial role when working with calculations involving exponential functions.

Change of Base Formula

The Change of Base Formula allows us to calculate logarithms with any base using natural logarithms. This is essential when calculators and tables are limited to specific bases like 10 or \(e\).

• The formula is \(\log_b a=\frac{\log_k a}{\log_k b}\), where \(k\) is the new base, usually \(e\) or 10.
• This formula lets us transform logarithms into a form that is easier to calculate.
• It is especially useful for converting between different logarithmic bases in complex equations.

By applying this formula, you can calculate \(\log_{10} 20\) as \(\frac{\ln 20}{\ln 10}\). This makes computations possible without always having to rely on base-specific tools or conversion tables since \(\ln\) is readily available.

Base e

The mathematical constant \(e\) is the base of natural logarithms and is an irrational number—one that cannot be expressed as a simple fraction. As a mathematical constant, it's almost everywhere in finance, physics, and biological modeling.

• It is approximately equal to 2.71828, though it has an infinite number of decimal places without repetition.
• Many natural processes are well-described using exponential functions with base \(e\), such as radioactive decay and population growth.
• It's the base for natural logarithms, making \(e\) central to calculus, especially in integration and differentiation of exponential functions.

Understanding \(e\) gives you a strong foundation in exploring exponential functions and their properties. Its unique characteristics allow mathematicians and scientists to describe and predict real-world phenomena with great accuracy.