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Chapter 3: Problem 56

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$-14+3 e^{x}=11$$

Short Answer

Expert verified
The solution to the equation \(-14 + 3e^x = 11\) is \(x 鈮� 2.197\) when rounded to three decimal places.

Step by step solution

01

Isolate the exponential term

Firstly, isolate the exponential term \(3e^x\) on one side of the equation. That can be achieved by adding 14 to both sides of the equation to eliminate \( -14\) on the left side. The new equation will now be \(3e^x = 11 + 14\)
02

Simplify the equation

After the equation gets simplified, it becomes \(3e^x = 25\) by adding 11 and 14 together.
03

Eliminate the constant in front of the base

Divide both sides of the equation by 3 to isolate \(e^x\). The equation now gets transitioned to \(e^x = 25/3\).
04

Apply natural logarithm

To solve for \(x\), utilize the natural logarithm (ln). The equation becomes \(ln(e^x) = ln(25/3)\). Because the natural logarithm and \(e\) are inverse functions, we can simplify our equation to \(x = ln(25/3)\).
05

Approximate to three decimal places

To find the approximate solution for \(x\), calculate the natural logarithm of \(25/3\) and round the result to three decimal places. The solution is \(x 鈮� 2.197 \)

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Most popular questions from this chapter

A logistic curve is also called a ________ curve.

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Find the exponential model \(y=a e^{b x}\) that fits the points shown in the graph or table. $$ \begin{array}{|l|l|l|} \hline x & 0 & 3 \\ \hline y & 1 & \frac{1}{4} \\ \hline \end{array} $$

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