Problem 82 - Calculus for Biology and Medicine

Question

Simplify the following expressions: (a) \(e^{3 \ln x}\) (b) \(e^{-\ln \left(x^{2}+1\right)}\) (c) \(e^{-2 \ln (1 / x)}\) (d) \(e^{-2 \ln x}\)

Step-by-Step Solution

Answer

(a) \(x^3\); (b) \(\frac{1}{x^2+1}\); (c) \(x^2\); (d) \(\frac{1}{x^2}\).

Step 1: Understand the Property of Exponentiation

The property \(e^{ ext{ln}(a)}=a\) and \(e^{b \, ext{ln}(a)}=a^b\) will be used to simplify each expression. This property helps convert expressions under the natural logarithm to their base forms.

Step 2: Simplify Expression (a)

Given \(e^{3 \ln x}\). Using the property from Step 1, this simplifies to \((x)^{3}=x^3\).

Step 3: Simplify Expression (b)

Given \(e^{-\ln(x^2+1)}\). Using the property, \(e^{-\ln(a)}=\frac{1}{a}\), it simplifies to \(\frac{1}{x^2+1}\).

Step 4: Simplify Expression (c)

Given \(e^{-2 \ln (1/x)}\), rewrite \(\ln(1/x)=-\ln x\). So, \(-2 \ln (1/x)= 2 \ln x\). Then, \(e^{2 \ln x} = (x)^2 = x^2\).

Step 5: Simplify Expression (d)

Given \(e^{-2 \ln x}\). Using the property, it simplifies to \((x)^{-2} = \frac{1}{x^2}\).

Key Concepts

Properties of Logarithms

Logarithms can be a bit confusing at first, but understanding their properties can make life much easier. A key property of logarithms is the power property. This property states that when you have a power inside a logarithm, you can bring it out in front. Mathematically, this is written as \(\ln(a^b) = b \ln(a)\). This means any exponent inside becomes a multiplier outside the log. Additionally, logarithms have an important relationship with their exponential counterparts. Particularly important for natural logarithms is the property \(e^{\ln(a)} = a\). Here, the exponential function and the natural log effectively "cancel" each other out, revealing the value underneath. This relationship is essential for simplifying expressions where exponential and logarithmic functions are mixed.

Simplifying Expressions

Simplifying expressions often involves using properties of exponentials and logarithms to rewrite them in easier forms. For instance, in the expression \(e^{3 \ln x}\), we can use the property of exponentiation with logarithms: \(e^{b\ln(a)} = a^b\). Applying this here, we successfully simplify the expression to \(x^3\).

Another interesting expression is \(e^{-\ln(x^2 + 1)}\). Using the property \(e^{-\ln(a)} = \frac{1}{a}\), we transform this into \(\frac{1}{x^2 + 1}\). The use of these properties can dramatically simplify otherwise complex-looking equations.

• Recognize properties of logs and exponents.
• Apply these properties step-by-step.
• Simplify to more manageable forms when possible.

Natural Logarithms

The natural logarithm, denoted as \(\ln\), is a logarithm to the base \(e\), where \(e\) is approximately 2.718. It is called "natural" because it often appears in real-world applications of exponential growth and decay. Because of its properties, the natural logarithm is incredibly useful in calculus and higher mathematics.

A natural logarithm has a unique ability to transform multiplications into additions, through the property \(\ln(ab) = \ln(a) + \ln(b)\). This capability is particularly useful in differential calculus and when dealing with exponential growth models.
When simplifying expressions involving natural logarithms, understanding the intertwining role of \(e\) and \(\ln\) can provide quick solutions. For example, if you stumble upon an expression like \(e^{2 \ln x}\), you can swiftly simplify it using exponentiation properties to \(x^2\). Clearly, the interplay between \(e\) and \(\ln\) reveals a charming simplicity!