Problem 66 - College Algebra and Calculus: An Applied Approach

Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln 4 x=1\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0.679\) to three decimal places.

Step 1: Convert the Logarithm to Exponential Form

Recall that the natural logarithm \(\ln a = b\) can be rewritten in exponential form as \(e^b = a\). Applying this property to the given equation \(\ln 4 x = 1\), the equation can be converted to \(4x = e^1\).

Step 2: Solve for x

To isolate \(x\), divide both sides of the equation by 4, which results in \(x = \frac{e^1}{4}\). Here, \(e^1\) is the mathematical constant \(e\) to the power of 1. Since anything to the power of 1 is itself, it simplifies to \(x = \frac{e}{4}\).

Step 3: Approximate to Three Decimal Places

Now we need to use a calculator to find the decimal approximation to the value of \(e\) divided by 4 to three decimal places. So, \(x = \frac{e}{4} = 0.679\) to three decimal places

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a powerful tool in mathematics, not just a mere symbol. At its core, \(\ln\) pertains to logarithms with base \(e\), approximately equal to 2.71828, which is known as Euler's number. Unlike base 10 logarithms, the natural logarithm is often used in calculations involving continuous growth, such as compound interest, population growth, and even in solving differential equations.
In any expression like \(\ln a = b\), the logarithm answers the question: To what power must \(e\) be raised, to obtain \(a\)? This unique attribute makes it particularly handy in converting into exponential forms, an essential step in solving logarithmic equations. Let's see how this transformation aids in solving the original equation \(\ln 4x = 1\) by transforming it into something more manageable.
The characteristics of the natural logarithm allow us to seamlessly convert into exponential form, which leads us into the next concept. This ability to interchange between forms is invaluable when tackling equations that initially appear complex.

Converting to Exponential Form

Transforming logarithmic equations into their exponential counterparts is an essential technique to make solving easier. By converting, we reveal another layer to the problem that is often simpler to handle. When you see \(\ln 4x = 1\), you're essentially asking: What power must \(e\) be raised to, to get \(4x\)?
Conveniently, the expression \(\ln a = b\) transforms into \(e^b = a\). For instance, in our problem, it's converted to \(4x = e^1\). By setting up the equation in this way, the natural logarithm simplifies and becomes a straightforward expression of \(x\).
Breaking it down further:

• Convert \(\ln 4x = 1\) to \(4x = e^1\).
• Recognize that \(e^1 = e\) since any number raised to the power of 1 is itself.

This shift from logarithmic to exponential form is not just a mathematical manipulation; it's a revelation that unveils the path to solving the equation efficiently.

Achieving Decimal Approximation

Once we have a simplified expression like \(x = \frac{e}{4}\), our goal shifts from pure algebra to practical approximation, essential in real-world applications where exact values are impractical. Decimal approximation allows us to represent real numbers, like the constant \(e\), in a finite number of digits.
Using a calculator or computational tool, we approximate \(e\) to about 2.71828. Then, plugging this into the expression gives \(\frac{2.71828}{4}\). This calculation results in \(0.67957\), which we round to three decimal places, resulting in \(0.679\).

• Use a calculator to evaluate \(\frac{e}{4}\).*
• Round the number to three decimal places.
• This renders our result suitable for quick comprehension and application in further problems or contexts.

Decimal approximations are vital for ensuring answers are concise and practical, especially in cases requiring precision such as scientific measurements.