Problem 52 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, solve for \(x\) or \(t\). $$ e^{\ln x^{2}}-9=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 3\) and \(x = -3\).

Step 1: Simplify the equation

Since \(e\) and \(\ln\) are inverse operations, \(e^{\ln x^{2}}\) can be simplified to \(x^{2}\). Therefore, the equation becomes \(x^{2} - 9 = 0\).

Step 2: Solve for \(x\)

The above equation can be solved by adding 9 to both sides of the equation and then taking the square root: \[(x^2) = 9 \Rightarrow x = \pm \sqrt{9}\].

Step 3: Find the values of \(x\)

So, the solutions are: \(x = +3\) and \(x= -3\).

Key Concepts

Natural Logarithm Properties

The natural logarithm, denoted as \(\ln\), is a fundamental concept in mathematics, particularly when dealing with exponential equations. One of its main properties is being the inverse of the exponential function \(e^x\). This means that \(\ln(e^x) = x\) and \(e^{\ln x} = x\), provided that \(x\) is positive. When solving exponential equations, understanding this property is essential because it provides a straightforward method to 'undo' the exponential function.

Another key property of the natural logarithm is its power rule: \(\ln(x^n) = n\cdot\ln(x)\). This property is valuable when you have an exponent inside the logarithm, as it allows you to move the exponent to the front as a multiplier, simplifying the expression and making the equation more manageable to solve.

These properties are not just abstract ideas; they are practical tools that aid in the transformation and simplification of complex equations into simpler forms that can easily be solved.

Inverse Operations

Inverse operations are mathematical actions that 'reverse' each other and are at the heart of solving many algebraic equations. For instance, addition and subtraction are inverse operations—adding 7 and then subtracting 7 from a number will lead you back to the original number. Similarly, multiplication and division are inverses. In the context of solving exponential equations, the exponential function \(e^x\) and the natural logarithm \(\ln\) are inverse operations.

In the given example, taking the natural logarithm of both sides could help solve an equation involving \(e^x\) by 'undoing' the exponential function. Conversely, if you have an equation involving a natural logarithm, you could exponentiate both sides using the base \(e\) to 'undo' the logarithm and simplify the equation.

Recognizing and correctly applying inverse operations are indispensable skills for solving algebraic equations, as these operations can eliminate complicated components of an equation, making it possible to isolate the variable of interest.

Square Roots

Square roots are a type of radical operation aimed at finding a number, which when multiplied by itself, gives the original number under the square root symbol. The square root of \(x\), which is written as \(\sqrt{x}\), asks the question: 'Which number times itself equals \(x\)?' In more formal terms, if \(y^2 = x\), then \(y = \sqrt{x}\).

An important aspect of square roots is that they have both positive and negative solutions, since both \(\sqrt{x}\) and \(\sqrt{x}\) squared will result in \(x\). This concept is crucial when solving equations where the variable is squared, as in the exercise provided. You would have to consider both the positive and negative square roots to ensure you find all possible solutions to the equation.

However, when dealing with real numbers, we primarily focus on the principal (non-negative) square root. Remember that every positive real number has two square roots, one positive and one negative. This is why the final step in solving the exercise equation results in two values for \(x\): \(+3\) and \(−3\).