Problem 46 - College Algebra

Question

Exercises \(45-52\) involve equations with natural logarithms. Solve each equation by isolating the natural logarithm and exponentiating both sides. Express the answer in terms of \(e\) Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln x=3$$

Step-by-Step Solution

Answer

The solution of the equation \(\ln x=3\) is \(x = e^3\), which is approximately 20.09

Step 1: Isolate the Natural Logarithm

The natural logarithm is already isolated in the equation \(\ln x = 3\)

Step 2: Exponentiate Both Sides to Remove the Natural Logarithm

In order to remove the natural logarithm, apply a base of \(e\) to both sides. This results in \(e^{\ln x} = e^3\). The base \(e\) and the natural logarithm cancel out on the left side, because \(e^{\ln x}\) essentially means 'the number \(e\) raised to the power that makes \(e\) equal (x), which is simply \(x\). Therefore we are left with \(x = e^3\)

Step 3: Express the Result in Terms of \(e\)

The left hand of the equation is x, and the right hand is expressed in terms of \(e\). So, \(x = e^3\)

Step 4: Approximate to Two Decimal Spaces

Approximate \(e^3\) to two decimal places using a calculator. The result approximately equals to 20.09

Key Concepts

Understanding Natural Logarithms

When we talk about natural logarithms, we're referring to logarithms with a very special base: the number e, which is approximately equal to 2.71828. This constant is foundational in mathematics and arises naturally in the study of growth and decay processes, among other areas. The natural logarithm of a number x is denoted by \(\ln x\) and is defined as the power to which e must be raised to obtain the number x.

For example, if we have \(\ln x = 3\), this is the same as saying that e raised to the third power equals x. Understanding this allows us to discern the true meaning behind the equation: we are looking for a number which, when e is used as a base and raised to it, gives us the original number x. It's a backbone concept for solving natural logarithm equations which will be expanded further with the process of exponentiation.

Exponentiating Logarithmic Equations

The process of exponentiating logarithmic equations is essentially 'undoing' a logarithm to get back to the value before it underwent the logarithmic operation. In simple terms, we are raising both sides of the equation to the power of a base, usually the base from which the log was derived.

In our example, to solve \(\ln x = 3\), we exponentiate both sides with the base e. This gives us \(e^{\ln x} = e^3\). The left side simplifies to x since the exponential function and the natural logarithm cancel each other out (\(e^{\ln x} = x\)). This wonderful property is called the inverse property of logarithms and is crucial for making these types of problems manageable.

Logarithmic Expressions

Now that you're familiar with the natural logarithm and the process of exponentiating, let's look closer at logarithmic expressions. A logarithmic expression involves the logarithm of a number or algebraic expression. It can be re-written and simplified using laws of logarithms, such as the product, quotient, and power rules.

When faced with a logarithmic equation, we sometimes need to manipulate these expressions to isolate the logarithm before we can exponentiate. For instance, if you have \(\ln(x+2) = a\), to isolate \(x\), you must first exponentiate, leading to \(x+2 = e^a\), then solve for \(x\) by subtracting 2 from both sides. Understanding how to handle these expressions by isolating and then removing the logarithm with exponentiation is a critical skill in algebra and higher mathematics.