Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 7 through 32, solve for \(x .\) $$ \ln x^{2}=3+\ln x $$

Step-by-Step Solution

Answer

The solution to the equation \( \ln(x^{2}) = 3 + \ln(x) \) is \( x = e^{3} \).

Step 1: Use Rule of Logarithms

Apply the rule of logarithms that says the logarithm of a power is the power times the logarithm, that is \( \ln(a^{b})=b \cdot \ln(a) \). Thus, our given problem \( \ln(x^{2})=3+\ln(x) \) simplifies to \( 2 \cdot \ln(x)=3+\ln(x) \).

Step 2: Rearrange to one side

We subtract \( \ln(x) \) from both sides of the equation to set it to 0 which gives us: \( 2 \cdot \ln(x) - \ln(x) - 3 = 0 \). Factor the \(\ln(x)\) term out to get \( \ln(x) \cdot (2 - 1) - 3 = 0 \). This simplifies to \( \ln(x) - 3 = 0 \).

Step 3: Isolate the logarithm

To isolate the logarithm statement, we add 3 to both sides of the equation which gives us \( \ln(x) = 3 \).

Step 4: Solve for \( x \)

Finally, we remove the natural logarithm by taking \( e \) to the power of both sides of the equation. This gives us \( e^{\ln(x)} = e^{3} \), which simplifies to \( x = e^{3} \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special logarithm with the base \( e \), where \( e \) is an irrational number approximately equal to 2.71828. It is called 'natural' because it naturally appears in many areas of mathematics, notably in the calculation of continuous growth or decay processes, like interest or radioactive decay. The natural logarithm of a number \( x \) is the power to which \( e \) must be raised to yield that number. For instance, when solving the equation \( \ln(x^2)=3+\ln(x) \) as in our original exercise, understanding that \( \ln \) denotes a logarithm to the base \( e \) provides the foundation for utilizing the rules of logarithms to isolate and solve for \( x \).

When you encounter an equation with a natural logarithm, it implies a relationship between an exponent and an exponentiated value, which brings into play other logarithmic properties and the special relationship between \( e \) and its logarithms.

Rule of Logarithms

Power Rule

One of the pivotal rules that come in handy when solving logarithmic equations is the 'Power Rule'. This rule states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number, mathematically given as \( \ln(a^b) = b \cdot \ln(a) \). In our exercise, this rule transformed the original equation \( \ln(x^{2}) \) into \( 2 \cdot \ln(x) \) simplifying the problem significantly.

Product and Quotient Rules

Other rules include the 'Product Rule' and the 'Quotient Rule', which help in decomposing logarithms of products or quotients into sums or differences of logarithms. These rules, while not directly used in our exercise, are equally critical when dealing with more complex equations involving logarithms of multiple terms.

• Product Rule: \( \ln(a \cdot b) = \ln(a) + \ln(b) \)
• Quotient Rule: \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \)

Exponential Functions

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. Represented as \( b^x \), this function grows or decays at a rate proportionate to its current value. The most unique and significant exponential function in mathematics is \( e^x \), where \( e \) is the base of the natural logarithm. It is particularly remarkable because \( e \) is the rate of growth shared by all continually growing processes.

When solving for \( x \) in the equation with a natural logarithm like \( \ln(x) = 3 \) from our problem, we use the property that \( e^\ln(x) = x \), because the function \( e^x \) and the natural logarithm \( \ln(x) \) are inverse operations. Taking \( e \) to the power of both sides allowed us to transform the logarithmic equation back into an exponential form, \( e^{\ln(x)} = e^3 \), and thus directly solve for \( x = e^3 \), providing a clear solution to our problem.