Problem 4 - Precalculus

Question

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$\log _{3,3} 3.3$$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Understand the Problem

We need to simplify the logarithmic expression \( \log_{3.3} 3.3 \) using properties of logarithms.

Step 2: Use the definition of Logarithm

Recall that the logarithm \( \log_b a \) means the power to which the base \( b \) must be raised to get \( a \). Mathematically, it is expressed as \( b^x = a \).

Step 3: Simplify the Logarithm

In this problem, the base and the number are the same: \( \log_{3.3} 3.3 \). According to the property of logarithms, \( \log_b b = 1 \) for any positive number \( b eq 1 \).

Step 4: Apply the Identity Property

Using the identity property of logarithms, since the base (3.3) and the number (3.3) are identical, \( \log_{3.3} 3.3 = 1 \).

Key Concepts

Definition of Logarithm

Logarithms can initially seem complex, but they are a fundamental concept in mathematics. A logarithm answers the question: "To what power must a certain number (the base) be raised, to produce a given number?" In formal terms, if you have a number base, denoted as \( b \), and want to know how many times to multiply it to get another number, \( a \), you are looking for the logarithm of \( a \) to the base \( b \). This is written as \( \log_b a \). To put it more simply, if \( b^x = a \), then \( \log_b a = x \). Logarithms help us solve equations where the variable is an exponent. For instance, if you know \( b^x = a \), you can express \( x \) as \( \log_b a \), making logs very handy in various calculations and real-world problems.

Identity Property of Logarithms

The identity property of logarithms is a fundamental and simple rule that helps in swiftly simplifying certain types of logarithmic expressions. It states that the logarithm of a number is equal to 1 when the base and the value are the same. Mathematically, this is expressed as \( \log_b b = 1 \). Consider you have \( b^1 = b \); this equation indicates that raising \( b \) to the power of one gives you \( b \) back, which neatly illustrates why the identity property is true. The property proves valuable whenever you encounter a logarithm where the base equals the number itself. It simplifies calculations, leading directly to an answer without further manipulation or computation. This property is part of the reason calculations involving logs can be done so efficiently.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves leveraging properties of logarithms, including the identity property, to transform complex logarithms into simpler forms. By doing so, one can often find more straightforward solutions to seemingly difficult problems.Here are steps or tricks you can apply:

• Look for situations where the base and the number are the same, as they can quickly simplify using the identity property.
• Utilize the definition of logarithms to understand what the logarithm is asking you to solve—for example, which power of the base equals the given number.

The main trick is recognizing these properties at play in a given expression. For the exercise \( \log_{3.3} 3.3 \), because the base and the number are identical, applying the identity property helps you directly simplify it to \( 1 \). This approach not only saves time but also reinforces understanding of how logarithms behave. Remember, practice helps these properties become more intuitive, smoothing out the learning process.