Problem 62 - Precalculus with Limits

Question

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. \(x=e^{-4}\)

Step-by-Step Solution

Answer

The value of the function \( g(x) = \ln x \) at \( x = e^{-4} \) is -4.

Step 1: Identify the value of x

The value of x for which we'll evaluate the function g(x) is \( x = e^{-4} \).

Step 2: Substitute x in the function

Now we substitute \( x = e^{-4} \) into the function \( g(x) = \ln x \) which gives us \( g(e^{-4}) = \ln(e^{-4}) \).

Step 3: Simplify the function

Recall that the natural logarithm function \( \ln \) can cancel out the exponential \( e \). This leaves us with \( \ln(e^{-4}) = -4 \).

Key Concepts

Understanding Exponential Functions

Exponential functions are a cornerstone in mathematics, particularly useful in representing growth and decay processes. An exponential function can be generally expressed as \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm, approximately 2.718. The parameter \( a \) defines the function's initial value, and \( b \) determines the rate of growth or decay.

Exponential functions exhibit the following characteristics:

• They have a constant percentage rate of change, meaning they grow or decay by a consistent proportion over equal intervals.
• The graph of an exponential function is smooth and continuous, never touching the x-axis.
• Both growth (\( b > 0 \)) and decay (\( b < 0 \)) functions are represented using exponential functions.

In our exercise, we deal with the expression \( e^{-4} \), which indicates an exponential decay due to the negative exponent. Understanding this helps when evaluating natural logarithms of such expressions.

Steps to Evaluate Logarithmic Expressions

Evaluating logarithmic expressions, like \( g(x) = \ln(e^{-4}) \), involves understanding the relationship between logarithms and exponents. The natural logarithm, \( \ln \), is the inverse function of the exponential function. This property enables us to simplify expressions efficiently.

When \( x = e^{-4} \), substituting it in \( g(x) \) means finding \( \ln(e^{-4}) \). The inherent property of logarithms and exponents helps to simplify this:

• The natural logarithm \( \ln(e^a) = a \), given that logarithms are the inverse of exponential functions.
• Therefore, \( \ln(e^{-4}) = -4 \), as the logarithm of \( e^{a} \) cancels out \( e \), leaving the exponent \( a \).

This process is simple yet powerful, allowing us to determine values without a calculator by recognizing patterns and using inherent properties of logarithmic expressions.

Exploring Logarithm Properties

Logarithm properties are essential for simplifying and evaluating expressions easily. Here are some key properties:

• Product Property: The logarithm of a product is the sum of the logarithms of the factors. \( \log_b(MN) = \log_bM + \log_bN \).
• Quotient Property: The logarithm of a quotient is the difference between the logarithms of the numerator and denominator. \( \log_b\frac{M}{N} = \log_bM - \log_bN \).
• Power Property: The logarithm of a power is the exponent times the logarithm of the base. \( \log_b(M^n) = n\log_bM \).
• Change of Base Formula: Allows you to change the logarithm to another base. \( \log_bM = \frac{\log_kM}{\log_kb} \).

A specific property to note is the power property, which directly applies to \( \ln \) as we have shown with \( \ln(e^{-4}) = -4 \). Understanding and applying these properties allow for effective and error-free evaluations of complex logarithmic expressions, simplifying math significantly.