Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-34: Solve the equation. $$ e^{2 \ln x} \equiv 9 $$

Step-by-Step Solution

Answer

The solution is \( x = 3 \).

Step 1: Convert Exponential to Power

The expression can be rewritten using the identity for exponential and logarithms: \( e^{a \, ext{ln}\, b} = b^a \). Hence,\[ e^{2 \, ext{ln}\, x} = x^2. \] So, rewrite the equation:\[ x^2 = 9. \]

Step 2: Solve for x Using Roots

Now, solve for \( x \) by taking the square root of both sides:\( x = \pm 3. \)

Step 3: Check Valid Solutions

Since the original logarithmic expression involves \( \ln(x) \), we must consider only the positive value of \( x \). Thus, the valid solution is \( x = 3 \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. These functions take the form \( a^x \), where \( a \) is a positive constant known as the base. One of the most common bases for exponential functions is Euler's number, \( e \), approximately equal to 2.718.
This number is unique because it naturally arises in various mathematical contexts, especially in calculus and different mathematical modeling. Exponential functions grow rapidly as the variable increases, making them useful in modeling situations involving growth or decay, such as population growth or radioactive decay.
When solving equations involving exponential functions, it often becomes necessary to use logarithmic functions, which are inverses of exponential functions, to make the variable more accessible and solvable.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. While exponential functions involve raising a base to a power, logarithmic functions reverse this process. They determine the power to which a base must be raised to produce a certain number.
For instance, the logarithmic function \( \log_b(x) \) answers the question, "To what power must \( b \) be raised, to equal \( x \)?" The natural logarithm, expressed as \( \ln(x) \), uses the base \( e \) and is particularly important in calculus and mathematical modeling.
The equation \( e^{\ln x} = x \) illustrates the essential inverse relationship between exponentials and logarithms: applying a logarithm to an exponential expression simplifies the equation, making it easier to solve. When solving exponential equations like in the example, logarithmic properties, such as \( e^{a \ln b} = b^a \), are vital to simplifying and finding the value of \( x \).

Equations

Equations are statements asserting the equality between two mathematical expressions. In algebra, equations often serve to represent and solve problems involving unknown variables.
The primary goal in solving an equation is to find the value of the variable that makes the equation true. Equations can range from simple linear equations, like \( x + 2 = 5 \), to more complex ones involving quadratic, cubic, or even exponential terms.
In the provided exercise, we work with an exponential equation \( e^{2 \ln x} = 9 \). Solving such equations typically requires converting the exponential expression into a simpler form if possible, as shown in the solution, before applying algebraic techniques. Understanding the nature of the variables and their interactions is key to finding an accurate solution.

Algebraic Solutions

Algebraic solutions involve finding the value of unknowns in equations using algebraic methods. These methods include operations like addition, subtraction, multiplication, division, and especially in some cases, taking roots or logarithms.
For the equation \( e^{2 \ln x} = 9 \), a crucial algebraic step is rewriting \( e^{2 \ln x} \) using properties of exponential and logarithmic functions. This transforms it into the equation \( x^2 = 9 \).
Solving \( x^2 = 9 \) involves applying the square root to both sides, leading to potential solutions \( x = 3 \) and \( x = -3 \). However, considering the original equation involves \( \ln(x) \), only the positive solution \( x = 3 \) is valid because the logarithm of a non-positive number is undefined in real numbers.
This step demonstrates the necessity of checking and confirming the solutions within the context of the initial equation.