Problem 22 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

No real solution, as \( e^{8x + 8} \) cannot be negative.

Step 1: Simplify the Equation

First, simplify the given equation by isolating the exponential term. Start by adding 5 to both sides of the equation: \( 7 e^{8x + 8} = -95 + 5 \). This simplifies to \( 7 e^{8x + 8} = -90 \). Then, divide both sides by 7 to isolate the exponential term: \( e^{8x + 8} = -rac{90}{7} \).

Step 2: Assess the Exponential Term

Notice that the isolated equation \( e^{8x + 8} = -rac{90}{7} \) has a negative right side. Since the exponential function \( e^y \) never yields negative values, as \( e^y > 0 \) for all real \( y \), this equation has no real solutions. The equation is inconsistent within the real number system.

Key Concepts

Logarithms

Logarithms are the inverse operations of exponential functions. They are useful for solving equations where the variable is an exponent. If you have an equation of the form \(a^x = b\), taking the logarithm of both sides can help to solve for \(x\). This process involves using properties of logarithms, such as the power rule: \(\log(a^x) = x\log(a)\).
Logarithms convert multiplicative relationships into additive ones, making them particularly handy when dealing with exponential equations. However, it's important to have a valid equation where both sides are positive since logarithms of non-positive numbers are undefined in the realm of real numbers.

Exponential Function

An exponential function, like \(e^x\), is one where the variable is an exponent. In these functions, the base is raised to the power of the variable. For instance, \(e^{8x + 8}\) is an exponential function with \(e\) as the base.
One critical aspect of exponential functions is that they are always positive for all real numbers. Mathematically, this means \(e^x > 0\) for any real \(x\). This property is vital because it helps identify whether a solution exists in a given equation. If you end up with a negative number on the other side of the equation, it indicates the absence of real solutions, as exponential functions cannot equal a negative number.

No Real Solutions

When solving equations, not every problem leads to a valid solution, especially when dealing with exponential functions. As shown in the original exercise, the equation ended with a negative number on one side: \(e^{8x + 8} = -\frac{90}{7}\).
It is crucial to remember that exponential functions cannot yield negative values, which automatically means there are no solutions within the real number system for this equation. These equations are considered inconsistent in the realm of real numbers. Recognizing this helps avoid unnecessary calculations and allows to conclude that no real solution exists.

Isolation of Terms

The first step in solving an exponential equation is isolating the exponential term. This makes it easier to work with later in the process. For example, in the original exercise, the equation needed to isolate \(e^{8x + 8}\) by first simplifying the left and right-hand sides.
To isolate an exponential term, typically you'll perform operations to remove additional coefficients and constants. In the given exercise, dividing both sides by 7 successfully isolated the exponential part. This step is crucial to set up the equation for solving by other methods, such as taking logarithms, if a real solution is possible.