Problem 30 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36\) , find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) 3 $$\ln e^{4}$$

Step-by-Step Solution

Answer

The exact value of the logarithmic expression \(3\ln(e^{4})\) is \(12\).

Step 1: Apply Scaling Property of Logarithms

The given expression is \(3\ln(e^{4})\). The property \(c\ln(x) = \ln(x^{c})\) can be applied. Therefore, it simplifies to \(\ln((e^{4})^{3})\).

Step 2: Simplify the Expression Inside Logarithm

The expression inside the logarithm is a power of a power. So, simplify it using the rule \((a^{m})^{n} = a^{mn}\). This simplifies to \(\ln(e^{12})\).

Step 3: Apply Natural Logarithm of Exponential Property

Now, apply the property of \(\ln(e^{a}) = a\) where a is the exponent of \(e\). Here, \(a\) is \(12\). So, \(\ln(e^{12})\) simplifies to \(12\).

Key Concepts

Scaling Property of Logarithms

The scaling property of logarithms is an essential tool in simplifying complex logarithmic expressions. When you encounter a term like \(c\ln(x)\), you can transform it into a more manageable form by employing this property: \(c\ln(x) = \ln(x^{c})\). This is particularly useful because it allows you to condense the expression by turning a multiplication outside of the logarithm into a power inside it.

• Imagine you have \(3\ln(e^{4})\). According to the scaling property, this becomes \(\ln((e^{4})^{3})\).
• Effectively, you have now redefined the entire problem as a single logarithm involving a power, which is often simpler to resolve, especially when dealing with exponential base powers.

The scaling property is instrumental in breaking down larger logarithmic computations into smaller, bite-sized components that are easier to compute by hand.

Natural Logarithm

Natural logarithms, represented as \(\ln(x)\), are logarithms with the base \(e\), where \(e\) is approximately equal to 2.71828. An easy way to think about natural logarithms is as the inverse of exponential functions with the base \(e\). This means that if you have \(\ln(e^{a})\), it simplifies directly to \(a\).

• This property stems from the relationship between exponential and logarithmic functions, making simplifications straightforward.
• In the expression \(\ln(e^{12})\), using this property, you can immediately deduce that the result is simply \(12\).

Natural logarithms are powerful in various fields such as finance and sciences due to their unique properties and the natural occurrence of \(e\) in continuous growth processes.

Exponential Functions

Exponential functions are functions of the form \(f(x) = a^{x}\), where a is a positive constant. They are vital in mathematics due to their rapid growth and prevalence in modeling real-world phenomena. When the base of the exponential function is \(e\), this function becomes particularly significant, especially relating to natural logarithms.

• In our exercise, \(e^{4}\) is used, and raising an exponential function to another power leads to multiplication of exponents, such as in \((e^{4})^{3} = e^{12}\). This is a fundamental property of exponents addressed in the simplification process.
• Exponential functions are especially powerful as they define how quantities grow over time, making them ideal for calculations involving continuous data, like population growth, radioactive decay, and compound interest.

Understanding exponential functions and their relationship with logarithms allows you to tackle complex mathematical problems easily and efficiently, knowing how to break down and manipulate these functions logically.