Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{2} 8=x $$

Step-by-Step Solution

Answer

The solution is \( x = 3 \).

Step 1: Identify the Format

The equation \( \log_2 8 = x \) is in the form \( \log_a b = c \). Here, \( a = 2 \), \( b = 8 \), and \( c = x \).

Step 2: Apply the Conversion Formula

The hint suggests using the conversion \( \log_a b = c \leftrightarrow a^c = b \). Apply it so that \( 2^x = 8 \).

Step 3: Solve the Exponential Equation

To solve \( 2^x = 8 \), express 8 as a power of 2. Since \( 8 = 2^3 \), we have \( 2^x = 2^3 \).

Step 4: Equate the Exponents

Since we have the same base, equate the exponents: \( x = 3 \).

Key Concepts

Understanding Logarithms

Logarithms are an essential concept in mathematics, and they are often used to simplify complex mathematical expressions. At its core, a logarithm answers the question: "To what power must the base be raised to produce a given number?" For instance, in the equation \( \log_2 8 = x \), the base is 2, and the logarithm asks what power of 2 gives you 8.
There are some key points to remember about logarithms:

• Base: The number you are repeatedly multiplying in a logarithmic equation, represented as \( a \) in \( \log_a b = c \).
• Result: The number you get after raising the base to a specific exponent, \( b \) in the equation.
• Exponent: The number to which the base is raised, shown as \( c \).

By understanding these components, solving problems involving logarithms becomes easier. Converting logarithmic equations to exponential form is a common strategy, as illustrated in the next section.

Equation Solving Basics

Equation solving is the heart of mathematics, allowing you to find unknown variables. Let's delve into solving the equation \( 2^x = 8 \), which was derived from our logarithmic problem. Here are the steps that help solve such equations:
First, express different numbers with the same base. In our example, rewrite 8 as a power of 2. We know \( 8 = 2^3 \). This conversion allows both sides of the equation \( 2^x = 2^3 \) to have the same base.
When the bases match, simply equate the exponents. Therefore, \( x = 3 \). This is a powerful method because it turns complicated exponential equations into simpler linear ones.
For more complex equations, remember these tips:

• Isolate the variable: Work towards having the unknown on one side of the equation.
• Simplify: Break down the equation into more manageable pieces.
• Check your solution: Substitute back to ensure the solution satisfies the original equation.

Knowing how to manipulate and solve equations is crucial for mathematical success.

Conversion Formulas: Logarithmic to Exponential

Conversion formulas play a vital role in simplifying and solving mathematical problems, especially those involving logarithms and exponentials. The conversion between logarithmic and exponential form is particularly helpful here.
Consider the formula \( \log_a b = c \leftrightarrow a^c = b \). This formula tells you that a logarithmic equation can be rewritten in exponential form, making it easier to solve for the unknown variable. In our example, \( \log_2 8 = x \) converts to \( 2^x = 8 \), which is simpler to handle.
Advantages of using conversion formulas include:

• Simplification: Turns complex logarithmic expressions into a more familiar exponential form.
• Clarity: Helps visualize and understand relationships between numbers.
• Versatility: These formulas are applicable in numerous mathematical scenarios, from pure algebra to complex calculus.

In essence, mastering conversion formulas opens the door to better leveraging the mathematical tools at your disposal, and ensures effective problem-solving approaches. Always look for opportunities to use these conversions, as they can often provide a clearer path to your solution.