Problem 23 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To evaluate a base-10 logarithm with a calculator, use the __ key. To evaluate the base-e logarithm, use the __ key.

Step-by-Step Solution

Answer

Use the "log" key for base-10 and the "ln" key for base-e logarithms.

Step 1: Identify the logarithms

There are different types of logarithms, such as base-10 logarithms and base-e logarithms. In this exercise, you need to identify which keys on a calculator are used to evaluate each type.

Step 2: Determine the key for base-10 logarithms

For base-10 logarithms, often referred to as common logarithms, you use the "log" key on a calculator. This key allows you to calculate the logarithm with a base of 10.

Step 3: Determine the key for base-e logarithms

Base-e logarithms are known as natural logarithms. To evaluate these, you use the "ln" key on a calculator, where "ln" stands for the natural logarithm with a base of Euler's number, e (approximately 2.718).

Key Concepts

Base-10 Logarithm

The base-10 logarithm, commonly known as the common logarithm, is a logarithm with a base of 10. When you see \ \( \log_{10}(x) \ \), it refers to the power to which 10 must be raised to obtain the number \ \( x \ \). In mathematical terms, if \ \( 10^y = x \ \), then \ \( \log_{10}(x) = y \ \). This type of logarithm is especially useful in science and engineering for dealing with quantities like speed or sound levels, which vary over large ranges.

• Common Example: \ \( \log_{10}(100) = 2 \ \), because \ \( 10^2 = 100 \ \).
• Notation: Often, the base-10 logarithm is simply written as \ \( \log(x) \ \) without explicitly showing the base "10".

Understanding and utilizing this concept helps when dealing with large numbers in a simplified form.

Natural Logarithm

The natural logarithm, denoted as \ \( \ln(x) \ \), is a logarithm with base \ \( e \ \), where \ \( e \ \) is an irrational and transcendental number approximately equal to 2.718. This type of logarithm is unique because it arises naturally in various mathematical contexts, particularly in calculus and growth calculations.
The natural logarithm is especially significant in scientific fields because it simplifies many expressions that involve exponential growth and decay.

• Common Example: \ \( \ln(e) = 1 \ \), since \ \( e^1 = e \ \).
• Properties: The natural logarithm is the inverse operation of taking an exponential function of \ \( e \ \).

Understanding \ \( \ln(x) \ \) is essential for calculus and for modeling exponential relationships in nature.

Calculator Keys

To solve logarithmic problems, most calculators come equip with specific keys dedicated to logarithm calculations. These keys are intuitive and labeled, making them easy to use.

• Log Key (\( \log \)): This button calculates base-10 logarithms or common logarithms. It is mainly used to simplify large numbers or equations that involve doubling or decibel calculations. To find \ \( \log_{10}(x) \ \), simply enter the number \ \( x \ \) followed by the \( \log \) key.
• Ln Key (\( \ln \)): This button is used for evaluating natural logarithms. When a computation involves growth patterns (like population models) or decay (like radioactive decay), you'll often need to use the \ \( \ln \) key.

Familiarity with these keys allows you to perform logarithmic calculations swiftly and accurately. Remember, practice using these keys to become comfortable with various logarithm types.