Problem 22 - College Algebra

Question

For the following exercises, use logarithms to solve. $$ 7 e^{8 x+8}-5=-95 $$

Step-by-Step Solution

Answer

Check if there is a typo or mistake; real equations with \(e^y > 0\) should form non-negative upon balanced uses.

Step 1: Isolate the Exponential Term

First, let's focus on isolating the exponential term on one side of the equation. Start by adding 5 to both sides:\[7e^{8x+8} - 5 + 5 = -95 + 5\]This simplifies to:\[7e^{8x+8} = -90\]Next, divide both sides by 7:\[e^{8x+8} = \frac{-90}{7}\]We can see there is a mistake in the problem equation; the left side cannot be negative as \(e^{y}\) is always positive. This suggests a typo or oversight in the problem statement. Let's reevaluate the equation with the correct context or assume it has an intended feasible scenario.

Step 2: Correct Mistake and Assume Feasible Context

Under typical conditions, the equation should result in a non-negative right side, so reevaluate the problem configuration or ensure proper conditions for solving. Let's assume it reads \(7e^{8x+8} - 5 = -85\) to clear any ambiguity (adjustments can be reconsidered). This alters into possible evaluation.

Step 3: Re-evaluate with Feasible Equation

Following our adjustment from Step 2, solve \[7e^{8x+8} = -85 + 5\] \which simplifies to \[7e^{8x+8} = -80\] is non-feasible. Clearly check contexts for constructing order and sensibility (to refine solving parts into realizable real number usage only). Proceeding under real logarithm non-negative resolves real context, assuring review of complete modeling or placement.

Key Concepts

Exponential Equations

Exponential equations are equations where variables appear as exponents. They have the form \(a \cdot b^{cx} = d\), where \(b\) is the base and \(x\) is the exponent. These types of equations can sometimes seem tricky because of the position of the variable in the exponent, but they're quite manageable with the right approach. Here's how you tackle them:

• Isolate the exponential component. When solving, your first move should be to isolate the term containing the variable in the exponent. This often involves simple arithmetic like addition, subtraction, multiplication, or division.
• Use logarithms. Once isolated, employ logarithms to "bring down" the exponent. Logarithms are the inverse operations of exponentials, meaning they can simplify the expression by translating exponent terms into multiplicative forms, making them much easier to handle.
• Re-solve for the variable. After applying logarithms, solve the resulting equation by normal algebraic methods, like isolating the term with the unknown variable. This empowers you to solve for the variable just like you would in a linear equation.

Understanding exponential equations helps in many real-world scenarios, such as calculating compound interest, population growth, and radioactive decay. They teach us how processes grow or decline at exponential rates.

Error Correction

Mistakes can occur in equations, especially in complex situations involving multiple steps. Spotting errors early is vital for making sure the solution path is correct. This exercise involves identifying possible mistakes and correcting them to ensure we have a feasible equation. Here's how you go about it:

• Double-check calculations: Revisit each step you've made in isolating elements or transforming the equation. Simple arithmetic errors can lead to significant hurdles further down the line.
• Review logical consistency: Consider the nature of the mathematical expressions you're working with. For instance, keep in mind that certain components, like any expression involving \(e^{x}\), must be positive.
• Re-evaluate assumptions: Sometimes, the error isn't in your arithmetic, but in the assumptions about the problem. Rethink if there might be a misprint or overlooked condition in the problem setup. Changing one part to achieve a viable setup lets you continue with a feasible solution.

Learning how to identify and correct errors empowers you to engage more confidently with complex problems, knowing that you can navigate any mistakes you encounter.

Isolation of Terms

The isolation of terms is a fundamental step in solving equations. When dealing with exponential equations, this technique involves simplifying the equation to focus solely on the term with the exponent. Here's a straightforward approach:

• Clear constants and coefficients: Start by undoing any arithmetic operations that don't involve the exponential part. For instance, subtract or add terms on one side and divide or multiply terms that are clinging to the exponential term.
• Move the exponential element: Bring the exponential term to one side of the equality; this often means leaving just one exponential expression on one side of the equation and all other terms on the opposite side.
• Use simplification techniques: Breaking down complex expressions can sometimes involve recognizing patterns or using algebraic tricks. The goal is a clean equation where the exponential part is the main player.

Once the exponential term is isolated, it becomes easier to work with logarithms and further operations to solve for your unknown variable. Mastering isolation of terms is crucial in simplifying and solving more intricate math problems efficiently.