Problem 61 - Precalculus : Building Concepts and Connections

Question

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log _{2} 8$$

Step-by-Step Solution

Answer

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

Step 1: Compare to the exponential expression

We can rewrite the expression \( \log_{2}(8) \) as \( 2^n=8 \) where n is what we need to find.

Step 2: Determine the value of n

Our task is to find the value of n that satisfies the equation \( 2^n=8 \). Since \( 2^3=8 \), that means n must be 3.

Step 3: Substitute back into original expression

Re-substitute n back into the original log expression to obtain \( \log_{2}(8) = 3 \).

Key Concepts

Exponential Expressions

Exponential expressions are mathematical expressions where a number, known as the base, is raised to a power or an exponent. The general form is expressed as \( a^n \), where \( a \) is the base and \( n \) is the exponent. The exponent indicates how many times the base is multiplied by itself.

• For example, in \( 3^2 \), the base is 3, and the exponent is 2, meaning 3 multiplied by itself, or \( 3 \times 3 = 9 \).
• They play a crucial role in solving logarithmic equations since one of the main purposes of logarithms is to find the exponent needed to obtain a specific number when given a base.

Understanding exponential expressions is essential when dealing with logarithms, as they often require converting between forms to simplify or solve equations. This concept allows us to grasp the relationship between numbers in logarithms more intuitively.

Simplifying Expressions

Simplifying expressions refers to the process of reducing an expression to its simplest or most concise form without changing its value. This often involves combining like terms, factoring, or converting expressions into different forms to make them more manageable or easier to interpret.

• With logarithms, this often means finding where the exponent form and logarithmic form can interconvert seamlessly, such as realizing that \( \log_{2}(8) \) simplifies directly to 3.
• Simplification ensures that we interpret complex calculations with ease, ensuring accuracy in logarithms and other expressions.

When you simplify, it is important to always ensure that the meaning or function of the original expression is preserved. This simplification process often helps when solving equations or when comparing different expressions to find similarities or differences.

Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm with an unknown variable. The structure of such equations typically looks like \( \log_b(y) = x \), which essentially asks "to what power must the base \( b \) be raised to produce \( y \)?"

• For example, \( \log_{2}(8) = 3 \) asks for the power to raise 2 to get 8. Since \( 2^3 = 8 \), the answer is 3.
• Logarithmic rules like the product, quotient, and power rules help in breaking down more complex logarithmic expressions.

These equations often involve re-expressing the equation in a different form or using base-changing formulas to solve for the unknown variable. Understanding logarithmic equations helps in many areas of mathematics, including exponential growth problems and scientific computations.