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Chapter 4: Problem 108

Solve each equation. Find the exact solutions. $$e^{3 x-4}=1$$

Short Answer

Expert verified
The exact solution to the equation \(e^{3x-4} = 1\) is \(x = \frac{4}{3}\).

Step by step solution

01

Understand the Problem

The given equation is an exponential equation of the form \(e^{3x-4} = 1\). We need to solve for \(x\) by finding the value that makes the equation true.
02

Isolate the Exponential Expression

Since \(e^{3x-4} = 1\), notice that any number raised to the power of 0 is 1. Therefore, \(e^{0} = 1\).
03

Set the Exponent Equal to Zero

Since \(e^{3x-4} = 1\), we can set the exponent equal to zero: \(3x - 4 = 0\).
04

Solve for x

Solve the equation \(3x - 4 = 0\) for \(x\).Add 4 to both sides to get \(3x = 4\).Divide both sides by 3 to get \(x = \frac{4}{3}\).
05

Verify the Solution

Substitute \(x = \frac{4}{3}\) back into the original equation to verify: \(e^{3 \frac{4}{3} - 4} = e^{4 - 4} = e^{0} = 1\), which is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving exponential equations
Exponential equations are equations where the variable appears in the exponent, such as in the equation \(e^{3x-4}=1\). To solve these, you need to take specific steps to simplify and isolate the variable. First, understand that the equation format here is exponential. For equations of the form \(a^{f(x)} = b\), where \(a\) and \(b\) are constants, and \(f(x)\) is an expression with the variable, our goal is to isolate \(x\). This requires some manipulation to break down the equation.
isolating the exponential term
To solve an equation like \(e^{3x-4}=1\), the first step is to isolate the exponential term. Notice that in this problem, the base of the exponent, \(e\), is a constant that represents Euler's number (approximately 2.71828). Recall that any number raised to the power of zero equals 1. So, \(e^0 = 1\). To connect this to our problem, we understand that for \(e^{3x-4}\) to equal 1, the exponent must be zero. Thus, set the exponent equal to zero: \(3x - 4 = 0\). This isolates the term with the variable and makes it easier to solve.
verifying solutions
After solving the isolated equation, the final important step is to verify your solution. Solve \(3x - 4 = 0\) by first adding 4 to both sides: \(3x = 4\). Then, divide both sides by 3 to get \(x = \frac{4}{3}\). To verify, substitute \(x = \frac{4}{3}\) back into the original equation: \(e^{3 \frac{4}{3}-4} = e^0 = 1\). Since the left-hand side simplifies to \(e^0\), which equals 1, the solution \(x=\frac{4}{3}\) is indeed correct. Verification ensures that no mistakes were made during the process, confirming the accuracy of the solution.

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Most popular questions from this chapter

Determine the domain and range of the function \(y=\log _{2}(x-1).\)

Solve each problem. Because of the Black Death, or plague, the only substantial period in recorded history when the earth's population was not increasing was from 1348 to \(1400 .\) During that period the world population decreased by about 100 million people. Use the exponential model \(P=P_{0} e^{r t}\) and the data from the accompanying table to find the annual growth rate for the period 1400 to 2000 . If the 100 million people had not been lost, then how many people would they have grown to in 600 years using the growth rate that you just found? $$\begin{array}{|c|c|}\hline \text { Year } & \begin{array}{c}\text { World } \\\\\text { Population }\end{array} \\\\\hline 1348 & 0.47 \times 10^{9} \\\1400 & 0.37 \times 10^{9} \\\1900 & 1.60 \times 10^{9} \\\2000 & 6.07 \times 10^{9} \\\\\hline\end{array}$$

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