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Chapter 3: Problem 29

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$e^{3 x}=12$$

Short Answer

Expert verified
The value of \(x\) is approximately \(0.792\).

Step by step solution

01

Isolate the exponential term

In this step, the exponential term is already isolated on one side of the equation. So the equation remains the same: \(e^{3x} = 12\).
02

Apply natural logarithm to both sides

In order to remove the base \(e\), apply the natural logarithm (ln) to both sides of the equation: \(ln(e^{3x}) = ln(12)\).
03

Simplify

Use the law of exponents which states that \(ln(a^{b}) = b*ln(a)\). It simplifies the left-hand side to \(3x*ln(e)\). Now, the natural logarithm of \(e\) is \(1\), so our equation becomes: \(3x = ln(12)\).
04

Solve for X

To solve for \(x\), divide both sides of the equation by \(3\): \(x = ln(12)/3\).
05

Approximate the Result

To get \(x\) to three decimal places, calculate the natural logarithm of \(12\) and then divide it by \(3\). Use a calculator for this.

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