Problem 57 - Algebra 2

Question

Mental Math Solve each equation. $$ \log _{8} 2=x $$

Step-by-Step Solution

Answer

The solution to the equation \( \log_{8} 2 = x \) is \(x = 1/3\).

Step 1: Understand Logarithmic equations

We start by understanding the logarithmic equation provided. Here the base is 8, and we want to find the exponent, which is x, such that \(8^x = 2\).

Step 2: Simplify the bases

We realize that the base 8 can be expressed as \(2^3\). So we can express the equation as \((2^3)^x = 2\), which simplifies further to \(2^(3x) = 2\).

Step 3: Compare the exponents

Since the bases on both sides of the equation are now the same (base 2), we can equate the exponents for a solution. So we have \(3x = 1\).

Step 4: Solve for x

We divide both sides of the equation by 3 to isolate x. That gives us \(x = 1/3\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to the power of an exponent. This concept helps to express repeated multiplication of a base number. For instance, in the expression \(8^x\), 8 is the base that is multiplied by itself \(x\) number of times.
Let's break it down further:

• When we say \(2^3\), it means \(2 \times 2 \times 2 = 8\).
• The result or solution of this operation is also called the power.
• It's a key operation needed to understand logarithms, which are essentially the inverse of exponentiation.

Understanding exponentiation helps students break down complex expressions by simplifying the bases, as seen in the exercise where \(8\) became \(2^3\). This simplification makes it easier to solve logarithmic equations.

Mental Math

Mental math refers to the practice of doing calculations in your head without using paper, a calculator, or any other aids. It's a useful skill, especially when it comes to understanding logarithms and exponential expressions.
Developing mental math skills can aid in:

• Quickly simplifying expressions by recognizing patterns, like realizing 8 is the same as \(2^3\).
• Easily comparing exponents when bases have been simplified to the same number.
• Building a more intuitive number sense, which is crucial when solving equations mentally.

Building competence in mental math enhances problem-solving abilities and reduces the dependence on technological aids. In the context of solving logarithmic equations, being able to convert bases and compare exponents mentally streamlines the process.

Logarithmic Equations

Logarithmic equations involve the concept of logarithms, the inverse operations of exponentiation. A logarithm answers the question: "To what exponent must the base be raised, to produce a given number?" For instance, in \(\log_{8} 2 = x\), we seek \(x\) such that \(8^x = 2\).
Let's explore how these equations work:

• Identify the base of the logarithm. In \(\log_{8} 2\), 8 is the base.
• Re-express the base, if possible, to see it in simpler terms. For example, transforming 8 into \(2^3\) simplifies problem-solving.
• Equating the exponents is the next step when both sides have the same base, making it easier to isolate \(x\).

By understanding logarithmic equations, students can deduce unknown exponents and solve the expressions step by step. This exercise effectively aligns the properties of exponents and logarithms, fostering a deeper appreciation for their inter-connected nature.