Problem 41 - Intermediate Algebra

Question

For Problems \(41-50\), solve each equation. $$ \log _{7} x=2 $$

Step-by-Step Solution

Answer

The solution is \( x = 49 \).

Step 1: Understand the Equation

The given equation is \( \log_{7} x = 2 \). This means we need to find a value of \( x \) such that the logarithm base 7 of \( x \) is equal to 2.

Step 2: Convert the Logarithmic Equation to Exponential Form

The logarithmic equation \( \log_{7} x = 2 \) can be rewritten in exponential form by using the definition of a logarithm. This gives us \( x = 7^2 \).

Step 3: Compute the Exponentiation

Calculate the value of \( 7^2 \). This means multiplying 7 by itself: \( 7 \times 7 = 49 \).

Step 4: Write the Solution

From the calculations, we have found that \( x = 49 \). This is the solution to the equation \( \log_{7} x = 2 \).

Key Concepts

Understanding Logarithmic Equations

In mathematics, logarithmic equations involve logarithms with a specified base, representing repeated multiplication. The equation \( \log_{7} x = 2 \) indicates that we are looking for the number \( x \) where the logarithm of \( x \) with base 7 equals 2. Here, the base 7 dictates the power to which it has to be raised to produce \( x \). Logarithmic equations are utilized to solve problems across various fields, helping to simplify expressions involving large numbers or solving equations where the variable is an exponent.

• Logarithms are the inverse operations of exponentials.
• The base of a logarithm must be a positive number not equal to one.
• An equation such as \( \log_{b} x = y \) implies \( b^y = x \).

Grasping these principles forms the foundation for effectively solving logarithmic equations, enhancing comprehension and mathematical problem-solving skills.

Explaining Exponential Form

Converting a logarithmic equation to its exponential form is a key step in solving for unknowns. For \( \log_{7} x = 2 \), rewriting in exponential form using the rule \( \log_{b} y = z \) converts to \( b^z = y \). Therefore, \( \log_{7} x = 2 \) becomes \( 7^2 = x \). This transformation reveals that the problem asks for 7 to be raised to the second power to find \( x \).

• Exponential form helps simplify calculations, making the solution clearer.
• Recognizing that logarithmic and exponential forms are interchangeable is crucial for understanding these mathematical concepts.
• In exponential form, the base, exponent, and result are clearly defined, aiding in immediate evaluation.

Converting to exponential form simplifies many complex logarithmic calculations and offers a straightforward path to determining the value of a variable like \( x \) in our initial equation.

Problem-Solving Steps for Logarithmic Equations

Solving logarithmic equations systematically involves several steps to ensure clarity and accuracy. For the equation \( \log_{7} x = 2 \) the steps include understanding, converting, and computing.

• **Understand the Equation**: Recognize that the logarithm expresses the power needed for the base 7 to equal \( x \).
• **Convert the Equation**: Transition the logarithmic equation to exponential form. For example, \( \log_{7} x = 2 \) becomes \( x = 7^2 \).
• **Compute the Exponentiation**: Calculate \( 7^2 \) to find \( x \). This means multiplying without reliance on logarithms, where \( 7 \times 7 = 49 \).
• **Write the Solution**: Formulate the final answer as \( x = 49 \), confirming the calculated value satisfies the original logarithmic equation.

Following these problem-solving steps allows for a structured approach, fostering confidence in mathematical reasoning and effective tackling of similar logarithmic equations.