Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\log _{3} 9$$.

Step-by-Step Solution

Answer

The exact value of \( \log ‌_{3} 9 \) is 2.

Step 1: Understanding the logarithmic expression

The expression \( \log ‌_{3} 9 \) represents the power to which the base 3 must be raised to obtain the number 9. In other words, it represents the equation \( 3^{x}=9 \). Our job is to find the value of \( x \).

Step 2: Finding the value

Recall that 9 can be expressed as \( 3^{2} \). Therefore, \( x \) is equal to 2 because 3 raised to the power of 2 gives us 9.

Step 3: Answer

Thus, \( \log _{3} 9 = 2 \) because 3 to the power of 2 equals 9.

Key Concepts

Exponents

Exponents are a fundamental mathematical concept used to represent repeated multiplication of the same number. When you see a number written like this: \(a^n\), it means that the base \(a\) is multiplied by itself \(n\) times.
So, \(3^2\) means \(3 \times 3\), which equals 9.Let's break down how this works:

• The base is the number that is being multiplied, in this case, 3.
• The exponent tells us how many times we multiply the base by itself. Here, it is 2.

This concept is crucial when working with logarithms, because it helps us reverse-engineer multiplication into simpler terms. Understanding exponents makes solving logarithmic equations much easier.

Base Conversion

Base conversion is key when working with logarithms because it allows you to simplify expressions by changing them into a more manageable form.When we talk about the base in the context of logarithms, we're actually referring to the number that is repeatedly multiplied to create a power, like in our exercise where the base is 3.
To solve \( \log_{3} 9 \), we need to express 9 as a power of 3. Recognizing that 9 is \(3^2\) is an example of base conversion. Understanding base conversion aids not just in solving equations directly but also offers insights into relationships between different numbers. Being good at base conversion means you'll be able to deconstruct and simplify many expressions.

Logarithmic Equations

Logarithmic equations are equations that involve the logarithm of a quantity. In simpler terms, they're used to find out how many times one number, the base, must be multiplied by itself to achieve another number.
For example, in the logarithmic equation \( \log_{3}{9} = x \), you're trying to find the value of \(x\) and express it as \(3^x = 9\).Here are some points to understand:

• Logarithmic equations can be re-written as exponents to simplify them.
• The base in logarithmic functions serves the same role as in exponential expressions.
• Understanding the conversion of expressions from logarithmic to exponential form is crucial in solving these equations.

Learning how to work with logarithmic equations empowers you to solve diverse real-world problems where exponential growth or decay is involved.