Problem 46 - Algebra A Combined Function

Question

Solve. $$ \log _{2} 8=x $$

Step-by-Step Solution

Answer

The value of \( x \) is 3.

Step 1: Understand the Problem

We need to find the value of \( x \) in the equation \( \log_2 8 = x \). This equation means we need to find the power \( x \) to which 2 must be raised to get 8.

Step 2: Recall the Definition of Logarithm

If \( \log_b a = x \), then \( b^x = a \). Here, \( b = 2 \), \( a = 8 \), and \( x \) is what we need to find. So we rewrite it as \( 2^x = 8 \).

Step 3: Express 8 as a Power of 2

Notice that \( 8 \) can be expressed as a power of \( 2 \). We have \( 8 = 2^3 \).

Step 4: Set the Exponents Equal

Since \( 2^x = 2^3 \), we can deduce that \( x = 3 \).

Step 5: Verify the Solution

Verify by substituting the value back: \( \log_2 8 = 3 \) means \( 2^3 = 8 \), which is correct.

Key Concepts

Power Equations

In mathematics, power equations often appear in the context of exponential and logarithmic functions. They involve expressions where a number, known as the base, is raised to a certain power, called the exponent. For example, in the equation \( b^x = a \), the base is \( b \) and the exponent is \( x \). Solving power equations typically requires finding which exponent satisfies the equation when the base is known.

• An important step in solving power equations like \( \log_2 8 = x \) is to rewrite the equation in exponential form as \( 2^x = 8 \).
• This approach transforms the logarithmic equation into a more familiar form of a power equation.
• Once we rewrite it, we can easily solve for the exponent \( x \) by expressing \( a \) as a power of \( b \).

Identifying and rewriting power equations utilizing equivalent exponential forms helps simplify and solve these equations efficiently.

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising a number, called the base, to the power of an exponent. In the equation \( 8 = 2^3 \), the base is 2 and the exponent is 3, indicating that 2 is multiplied by itself 3 times to reach 8. This concept is essential for understanding how exponential growth or decay occurs throughout various natural and scientific phenomena.

• Exponentiation allows us to easily express large numbers or transformations with repeated multiplication.
• It plays a crucial role in the laws of logarithms, since logarithms are essentially the inverse of exponentiation.
• Being comfortable with reducing numbers to their base exponential forms is key in working with logarithmic equations.

A clear grasp of exponentiation will enable you to effortlessly manage calculations involving exponential growth, powers of numbers, and their corresponding logarithmic forms.

Logarithmic Functions

Logarithmic functions are an essential part of algebra and calculus, relating to the concept of inverse operations of exponentiation. In simple terms, a logarithm tells us the exponent or power to which we must raise a base number to produce a given number. For instance, in \( \log_2 8 = x \), the logarithm indicates that we need to find what power 2 should be raised to get 8.

• Logarithms simplify complex multiplicative processes into manageable addition tasks, greatly easing calculations.
• The basic form of a logarithm \( \log_b a = x \) can be reinterpreted as \( b^x = a \), connecting it back to exponential equations.
• When solving logarithmic functions, converting them to exponential form can often provide more clarity and insight.

Understanding logarithmic functions is crucial for solving equations where the unknown is an exponent and for applications in real-world problems that showcase exponential behavior.