Problem 61 - Precalculus with Limits

Question

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. \(x=e^{5}\)

Step-by-Step Solution

Answer

The evaluated value is \(5\).

Step 1: Understanding the function

The function given is \(g(x)=\ln x\). As per the property of logarithms, \(\ln(e^{x})=x\), the logarithm to the base \(e\) of \(e^{x}\) is \(x\).

Step 2: Substituting the value of \(x\)

The value of \(x\) given is \(e^{5}\). Substituting \(x=e^{5}\) in the function \(g(x)\) we get \(g(e^{5})\).

Step 3: Evaluating the function

Using our property of logarithms, we can simplify \(g(e^{5})\). We know that \(\ln(e^{5})=5\), therefore \(g(e^{5})=5\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In many cases, this constant base is the number \(e\), which is approximately equal to 2.71828. The notation for an exponential function usually looks like \(a^{x}\), where \(a\) is the base and \(x\) is the exponent.

• Exponential functions are essential for modeling growth and decay, such as population growth, radioactive decay, and interest calculations.
• In the context of the given exercise, \(e^{5}\) is an exponential expression where the base \(e\) is raised to the power of 5.
• An exponential function grows very quickly as the exponent increases.

Understanding these functions is crucial because their inverse is the logarithmic functions, which we'll discuss next.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. While exponential functions involve raising a base to an exponent, logarithmic functions find which power the base must be raised to obtain a certain number. The natural logarithm, written as \(\ln x\), is common in mathematics, specifically using the base \(e\). The function \(\ln x\) tells you the power that \(e\) must be raised to produce \(x\).

• If \(a^{b}=c\), then \(\log_{a}c=b\). For natural logarithms, it means that if \(e^{y}=x\), then \(\ln x=y\).
• In practical applications, logarithmic functions help solve equations where the unknown is in an exponent.
• The relationship between exponential and logarithmic functions is key to simplifying expressions, such as \(g(e^{5})=5\) in our exercise.

Logarithms can seem tricky at first, but they are incredibly useful for inverse calculations involving exponentials.

Properties of Logarithms

The properties of logarithms help simplify and solve various mathematical problems. Understanding these properties is essential for efficiently working with logarithmic expressions. Some fundamental properties include:

• Product Property: \(\ln(ab) = \ln a + \ln b\). It shows that the log of a product is the sum of the logs.
• Quotient Property: \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\). It indicates that the log of a quotient is the difference of the logs.
• Power Property: \(\ln(a^b) = b \ln a\). This property states that the log of a power is the exponent times the log of the base.

In the exercise, we apply the property \(\ln(e^{x}) = x\), which simplifies \(\ln(e^{5}) = 5\). This particular property shows why logarithms and exponentials are inverse operations, simplifying expressions efficiently. Understanding these properties can make complex calculations much more manageable.