Problem 20 - A Graphical Approach to College Algebra

Question

Solve each equation. Give the exact answer. $$\log _{3} 81=x$$

Step-by-Step Solution

Answer

x = 4

Step 1: Identify the Relationship

Understand that the equation \(\log_{3} 81 = x\) means "which power must 3 be raised to in order to get 81?" This relation is key to solving logarithmic equations.

Step 2: Rewrite as an Exponential Equation

Convert the logarithmic equation \(\log_{3} 81 = x\) to its exponential form: \(3^x = 81\). This step makes it easier to compare and solve for \(x\).

Step 3: Recognize the Powers of 3

Identify that 81 is a known power of 3. Recall that \(3^4 = 81\). Therefore, the equation \(3^x = 81\) implies \(x = 4\).

Step 4: Check the Answer

Verify the solution by evaluating \(\log_{3} 81\): Since \(3^4 = 81\), these two equations are consistent, confirming that the solution \(x = 4\) is correct.

Key Concepts

Exponential Equations

Exponential equations are equations where the unknown variable appears as an exponent. They're central to various real-world applications like population growth and financial modeling. To solve these equations, it's helpful to express them in a way where the bases on both sides of the equation can be compared.Let's break down an example:- Consider the expression \(3^x = 81\). Here, 3 is the base, and \(x\) is the unknown exponent.- The strategy is to express 81 as a power of 3, which allows you to compare the exponents directly. Knowing that \(81 = 3^4\) leads to the conclusion that \(x = 4\).In general, solving exponential equations involves:- **Identifying the base:** What number is being raised to a power?- **Recognizing common powers:** Can the numbers involved be expressed as similar bases?- **Solving for the exponent:** Once bases match, compare and solve for the unknown exponent.This approach helps simplify and solve exponential equations by turning them into a more manageable form.

Powers of Numbers

Understanding the powers of numbers is crucial when dealing with exponential and logarithmic equations. Powers tell us how many times to multiply a number by itself. For example, \(3^4\) means 3 multiplied by itself four times: \(3 \times 3 \times 3 \times 3 = 81\).Here are some pointers for handling powers of numbers:- **Common powers:** Be familiar with common powers, especially small numbers like 2, 3, and 5. They often appear in equations, so knowing these by heart can be helpful.- **Factorization:** Break down numbers into their prime factorization to identify powers. For example, 81 is the product of multiplying four 3s (i.e., \(3^4\)).- **Recognizing patterns:** Recognize patterns in powers that simplify finding solutions.Mastering the powers of numbers simplifies working with complex exponential equations. It allows you to recognize relationships and solve equations efficiently.

Solving Equations

The process of solving equations is about finding the value of the unknown variable that makes the equation true. Here, we delve into techniques specifically useful for solving exponential and logarithmic equations.**Steps to Solve Exponential and Logarithmic Equations:***

• **Interpreting the equation:** Understand what the equation is asking. For logarithmic equations, this involves converting them into exponential form for easier manipulation.
• **Convert forms:** Change the forms between logarithmic and exponential. For example, \(\log_{3} 81 = x\) becomes \(3^x = 81\), making it easier to see the relationship.
• **Equalize the bases:** Where possible, express both sides of the equation with the same base. This allows the equation to be solved by equating the exponents.
• **Solve for the variable:** Once in easier form, solving becomes a matter of basic algebraic manipulation, like setting exponents equal to each other.

Mastering these steps not only aids in solving equations but also strengthens overall algebraic skills.