Problem 2 - Precalculus Functions and Graphs

Question

Fill in the blank. A base 10 logarithm is a(n) __________ logarithm.

Step-by-Step Solution

Answer

common

Step 1: Understanding Logarithms

Recall that a logarithm answers the question: 'To what exponent must a specific base be raised, to get a particular number?'

Step 2: Identifying the Base

In this problem, the base is mentioned as 10, which is a common base for logarithms.

Step 3: Common Terminology

Logarithms with base 10 have a special name in mathematical terminology. They are known as 'common logarithms'.

Step 4: Fill in the Blank

Based on the previous step, the blank in the sentence 'A base 10 logarithm is a(n) __________ logarithm' should be filled with the term 'common'.

Key Concepts

base 10 logarithm

A base 10 logarithm is a logarithm that uses 10 as its base. In other words, it answers the question, 'To what power must we raise 10 to get a given number?' For example, if you wanted to find the base 10 logarithm of 100, you would be asking, 'What power must I raise 10 to get 100?' The answer is 2 because 10 raised to the power of 2 is 100: \[ \log_{10}(100) = 2 \] Base 10 logarithms are very common in scientific calculations and everyday applications. This is why they are often referred to as common logarithms.

logarithms

Logarithms are a fundamental concept in mathematics that tell us how many times one number, called the base, needs to be multiplied by itself to produce another number. The general form is written as \[ \log_b(a) = c \] where

• \(b\) is the base,
• \(a\) is the number you're taking the logarithm of,
• and \(c\) is the exponent to which you must raise the base to get \(a\).

Logarithms are used in many areas of science and engineering to simplify complex calculations, such as dealing with large numbers or exponential growth and decay. For example, in finance, logarithms can be used to calculate compound interest, and in physics, they can help describe the intensity of sound or light.

mathematical terminology

Understanding mathematical terminology is crucial for mastering concepts like logarithms. Terms such as 'base', 'exponent', and 'logarithm' must be clearly understood.

• 'Base': The number that is raised to a power.
• 'Exponent': The power to which the base number is raised.
• 'Logarithm': The exponent to which the base must be raised to produce a given number.

In the given problem, a base 10 logarithm is called a 'common logarithm', highlighting its frequent use. These terminologies help in deciphering and understanding complex mathematical expressions and solutions effectively.