Problem 26 - Precalculus with Limits

Question

Use the properties of logarithms to simplify the expression. \(\log _{3.2} 1\)

Step-by-Step Solution

Answer

The simplified form of \(\log _{3.2} 1\) is 0.

Step 1: Apply the properties of Logarithms

Recalling the property of logarithms, we have that \(\log _{a} 1 = 0,\) where \(a\) is any base. So, this property can be applied straightly to the expression.

Step 2: Direct Substitution

Substituting the values from our exercise into the formula, we get \(\log _{3.2} 1 = 0\).

Key Concepts

Logarithmic Expressions

A logarithmic expression is a mathematical concept where a logarithm is used to represent the power to which a number, known as the base, must be raised to produce another number. Simply put, if we have an expression like \( \log_{a} b \), it indicates "what power must \( a \) be raised to, to get \( b \)?" This question is addressed by the logarithm.
Logarithmic expressions are crucial in simplifying complex multiplicative relationships, especially in the fields of science and engineering. They can be used extensively to solve exponential equations as well.

• Key parts: Logarithms involve a base, and a number for which we are trying to find the exponent.
• Notation: \( \log_{a} b \), where \( a \) is the base and \( b \) is the number in question.

This type of expression helps in reducing complex operations into simpler forms, thus making it easier to handle large-scale computations.

Base of Logarithm

The base of a logarithm is a critical component, as it specifies the number that we are repeatedly multiplying. In a logarithmic expression like \( \log_{a} b \), \( a \) is known as the base.
The base plays an important role in determining the value of the logarithm. Different bases can lead to different interpretations and values. For example, the binary system uses a base of 2, the common logarithm uses base 10, and the natural logarithm uses base \( e \), approximately equal to 2.718.

• Why is it important? Because it determines the number of times the multiplication occurs.
• Flexibility: You can change the base using logarithmic properties, such as the change of base formula: \( \log_{b} c = \frac{\log_{a} c}{\log_{a} b} \).

Understanding the base lets us solve logarithmic problems by applying the correct logic based on the type of base used.

Simplifying Logarithms

Simplifying logarithms involves reducing logarithmic expressions into more manageable numbers or forms. One of the most useful properties in this process is the rule that states the logarithm of 1 with any base is always 0. This means \( \log_{a} 1 = 0 \), regardless of what \( a \) is.
Why is this helpful? It allows for much simpler calculations by eliminating more complex steps. When tackling problems, it's important to know:

• Basic Properties: In addition to \( \log_{a} 1 = 0 \), remember \( \log_{a} a = 1 \) and \( \log_{a} (a^b) = b \).
• Combination and decomposition: Use properties to combine or break down logs, such as \( \log_{a}(xy) = \log_{a}x + \log_{a}y \).

Be familiar with these properties as they can drastically reduce the amount of computation required and simplify the broader understanding of log expressions.