Problem 303 - Precalculus

Question

Use logarithms to solve. \(e^{r+10}-10=-42\)

Step-by-Step Solution

Answer

No real solutions.

Step 1: Isolate the Exponential Expression

Start by isolating the term with the exponent. Add 10 to both sides of the equation to eliminate the '-10' on the left side:\[ e^{r+10} - 10 + 10 = -42 + 10 \]This simplifies to:\[ e^{r+10} = -32 \]However, this equation doesn't hold as real numbers because the exponential function is always positive, thus we can't have a negative result on the right side.

Step 2: Analyze the Equation's Validity

Since exponential functions of the form \(e^x\) are always positive, the equation \(e^{r+10} = -32\) is not valid. Thus, there are no real solutions to \(e^{r+10} - 10 = -42\).

Key Concepts

Exponential Functions

Exponential functions are a fundamental mathematical concept used in various fields such as physics, biology, and finance. These functions are expressed in the form of \( f(x) = a^x \), where \( a \) is a positive constant greater than zero, and \( x \) is any real number. A defining characteristic of exponential functions is their rapid growth or decay.
If \( a > 1 \), the function represents exponential growth, meaning the function's values increase as \( x \) increases. Conversely, if \( 0 < a < 1 \), the function describes exponential decay, where values decrease as \( x \) increases. In this context, the exponential function \( e^x \), where \( e \) approximately equals 2.718, is often used due to its natural properties in calculus and its occurrence in real-world applications.
An essential feature of these functions is that their range is always positive; they never produce negative values. This is crucial when solving equations involving exponential expressions, as it provides insight into the possible solutions and potential restrictions of the equation.

Equations

Equations are mathematical expressions that assert the equality of two expressions by using an equals sign "=". When dealing with exponential equations like \(e^{r+10}-10=-42\), it's crucial to isolate the exponential part of the equation first to simplify the problem. For the given equation, adding 10 to both sides helps to isolate the exponential expression:

• This results in \( e^{r+10} = -32 \).
• However, as exponential functions yield only positive values, this equation indicates a problem.

The left side, \(e^{r+10}\), represents an exponential function, which cannot equal \(-32\) because it's negative.
In general, exponential equations can often be solved using logarithms to "undo" the exponentiation, if the right side of the equation is positive. However, when faced with a negative on the right side, the equation has no real solution due to the characteristics of exponential functions.

Real Solutions

Real solutions refer to solutions of an equation that belong to the set of real numbers, which include all rational and irrational numbers. When solving equations, the goal is often to find all possible real solutions. However, not all equations will yield real solutions, which occurs when the equation’s conditions or constraints cannot be satisfied by any real number.
In the case of the equation \(e^{r+10} - 10 = -42\), attempting to solve for \(r\) would lead to an impossibility due to the nature of exponential functions, as noted earlier. Since the exponential function is always positive and the right side of the isolated equation is negative, no real number \(r\) can satisfy the equation.
Thus, when analyzing mathematical problems, it is critical to understand the properties of the functions and the feasibility of obtaining real solutions. This understanding allows mathematicians and students alike to determine if further efforts to find solutions are necessary or if a conclusion, like "no real solution," is appropriate.