Problem 38 Solve. Where appropriate, includ... [FREE SOLUTION]

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Chapter 12: Problem 38

Solve. Where appropriate, include approximations to three decimal places. $$ \ln (3 x)=2 $$

Short Answer

Expert verified

x 鈮� 2.463

Step by step solution

Identify the given equation

The given equation is $\ln (3x) = 2$. The objective is to solve for x.

Isolate the logarithmic expression

The logarithmic expression $\ln(3x)$ is already isolated on the left side of the equation.

Exponentiate both sides of the equation

To solve for x, exponentiate both sides to eliminate the natural logarithm. \[ e^{\ln(3x)} = e^2 \]Since $e^{\ln y} = y$, it simplifies to: \[ 3x = e^2 \]

Solve for x

Divide both sides by 3 to isolate x: \[ x = \frac{e^2}{3} \]

Evaluate the exponential expression

Use the approximate value of e (Euler's number) which is about 2.718. Thus, \[ e^2 \approx 2.718^2 \approx 7.389 \]Now, compute $\frac{7.389}{3}$: \[ x \approx \frac{7.389}{3} \approx 2.463 \]

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Key Concepts

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

Natural Logarithm

Exponentiation is a mathematical operation involving the raising of a base, e, to the power of an exponent. In our equation, we need to 'undo' the natural logarithm by exponentiating both sides. This utilizes the property $e^{\ln y}=y$.
So, by exponentiating both sides of $\ln (3x) = 2$, we get:
\[ e^{\ln (3x)} = e^2\]
Which simplifies to $3x = e^2$.
This step is crucial in solving for $x$.

Euler's Number

Euler's number, denoted as $e$, s a mathematical constant approximately equal to 2.718. It is an irrational number, meaning it cannot be expressed as a simple fraction. This number is fundamental in mathematics, particularly in the fields of calculus and complex analysis.
In the context of our problem, Euler's number helps in converting the natural logarithm equation to an exponential form. Specifically, we use its property in the expression \[ 3x = e^2\] to isolate and solve for $x$. Here, knowing that $e^2 \approx 7.389$, helps us find a numerical solution.

Approximation

In this exercise, we encountered the need to find an approximate numerical value for Euler's number raised to the power of 2, $e^2$. Given that $e \approx 2.718$, we compute:
\[e^2\approx 2.718^2\approx 7.389\].
This approximate value helps us in further calculations. Finally, we divide by 3 to solve for $x$, leading us to:
\[ x \approx \frac{7.389}{3}\ \approx 2.463\].
This approximation is useful when an exact value isn't necessary, especially in homework or practical applications.

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91影视

Solve. Where appropriate, include approximations to three decimal places. $$ \ln (3 x)=2 $$

Short Answer

Step by step solution

Identify the given equation

Isolate the logarithmic expression

Exponentiate both sides of the equation

Solve for x

Evaluate the exponential expression

Key Concepts

Natural Logarithm

Euler's Number

Approximation

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