Problem 29 - Algebra A Combined Function

Question

Find the value of each logarithmic expression. $$ \log _{2} 8 $$

Step-by-Step Solution

Answer

The value of \( \log_{2} 8 \) is 3.

Step 1: Understand the Logarithmic Expression

The expression \( \log_{2} 8 \) asks for the power to which 2 must be raised to equal 8.

Step 2: Rewrite the Expression as an Exponential Equation

Set the expression as an equation: \( 2^x = 8 \). This means we are looking for the value of \( x \) that makes this equation true.

Step 3: Solve the Exponential Equation

Recognize that \( 8 \) can be rewritten as \( 2^3 \), since \( 2 \times 2 \times 2 = 8 \). Thus, \( 2^x = 2^3 \).

Step 4: Equate the Exponents

Since the bases are the same, you can equate the exponents: \( x = 3 \).

Step 5: Verify the Solution

Check that \( 2^3 = 8 \). Since this is correct, \( \log_{2} 8 = 3 \).

Key Concepts

Exponential Equations

An exponential equation involves variables in the exponent. When you have an equation like \(2^x = 8\), the goal is to find the value of \(x\). This means that you're trying to determine what power you have to raise the base (in this case, 2) to, in order to get the result (8).
To solve exponential equations, it helps to express both sides of the equation with the same base. Once this is achieved, you can simply equate the exponents. For example:

• Identify the base of the exponential equation.
• Rewrite the number on the other side of the equation using this base, if possible.
• Equate the exponents since the bases are the same.

In this exercise, we solved \(2^x = 8\) by rewriting 8 as \(2^3\), allowing us to see clearly that \(x\) must equal 3. This technique simplifies solving such equations and relies on recognizing number patterns or converting to a common base.

Exponents

Exponents describe the number of times a base is multiplied by itself. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent. This means: "2 multiplied by itself 3 times."
Understanding the role of exponents is crucial when dealing with exponential equations or logarithmic expressions. Here are some key points:

• Exponents are a shorthand for repeated multiplication.
• Exponential growth involves increasing values rapidly due to repeated multiplication.
• When the base is greater than 1, larger exponents lead to larger numbers.

Exponents can be manipulated through several rules:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Zero Exponent: \(a^0 = 1\)

Recognizing these properties can simplify complex calculations and make evaluating expressions more efficient.

Evaluating Logarithms

Evaluating logarithms involves finding what power you need to raise a base to get a certain number. The expression \(\log_{b} a\) means: "to what power must \(b\) be raised to equal \(a\)?"
This concept is rooted in the fact that logarithms are the inverses of exponents. Here’s how to effectively evaluate logarithms:

• Convert the log expression into an exponential form, for example, \(\log_{2} 8 = x\) becomes \(2^x = 8\).
• Simplify the expression by rewriting the number as a power of the base, if possible.
• Find the value of the exponent, which is your log answer.

Logarithms transform multiplication into addition and division into subtraction, making them useful in complex calculations. For instance, they help us solve exponential equations by transforming them into manageable expressions.
It's crucial to understand that evaluating logarithms is all about finding the right exponent that makes the equation true, as seen in the solution \(\log_{2} 8 = 3\), where 2 raised to the power of 3 equals 8.