Problem 55 - Precalculus with Limits

Question

Write the exponential equation in logarithmic form. \(e^{-0.9}=0.406 \ldots\)

Step-by-Step Solution

Answer

The logarithmic form of the equation \(e^{-0.9}=0.406\) is \(\ln 0.406 = -0.9\).

Step 1: Understanding the equation

The given equation is \(e^{-0.9}=0.406\). In this equation, \(e\) is the base, \(-0.9\) is the power or exponent, and \(0.406\) is the result of the exponentiation.

Step 2: Applying the conversion rule

To convert an exponential equation into its equivalent logarithmic form, the base of the exponential term becomes the base of the logarithm. The result of the exponentiation becomes the argument of the logarithm, and the exponent becomes the solution of the logarithm. Therefore, the equivalent logarithmic equation of an exponential equation of the form \(b^y = x\) is \(\log_b x = y\).

Step 3: Writing the logarithmic form of the equation

Using the rules from step 2, the given equation \(e^{-0.9}=0.406\) can be rewritten in logarithmic form as \(\log_e 0.406 = -0.9\). The base \(e\) of the natural logarithm is usually not written explicitly. Hence, the final equivalent logarithmic form of the equation is \(\ln 0.406 = -0.9\).

Key Concepts

Exponential Functions

In mathematical terms, an exponential function is a type of mathematical function where a constant base is raised to a variable exponent. This means the variable is in the exponent rather than the base, setting exponential functions apart from other mathematical expressions. The general form of an exponential function is expressed as \(b^x\), where \(b\) is a positive real number not equal to one, known as the base, and \(x\) is the exponent.
This type of function is used to describe growth and decay processes, such as population growth, radioactive decay, and compound interest.

• Key Property: An important property of exponential functions is that their rate of growth is proportional to their value.
• Example: The function \(2^x\) doubles for each increase of 1 in \(x\), showcasing the rapid growth characteristic of such functions.

Regardless of the specific context, these functions are vital in helping us describe processes in which quantities increase or decrease rapidly.

Logarithmic Functions

Logarithmic functions serve as the inverse functions of exponential functions. While exponential functions involve a base raised to a power, logarithmic functions tell us the power to which a base must be raised to obtain a given number. The general form is \(\log_b x = y\), where \(b^y = x\). In this equation:

• \(b\) is the base of the logarithm, which is equivalent to the base of the corresponding exponential function.
• \(x\) is the argument, representing the number we are seeking to find a log for.
• \(y\) is the logarithm itself, showing the exponent.

Logarithms simplify many calculations by turning multiplicative processes into additive ones. This characteristic is particularly useful in solving equations involving exponential decay or growth. Special types of logarithms include the natural logarithm (base \(e\)) and the common logarithm (base 10).

Conversion Between Forms

Converting between exponential and logarithmic forms is crucial for solving equations in both formats. Understanding the relationship between these two forms can simplify complex mathematical problems. The conversion is based on the equivalence \(b^y = x\) in exponential form and \(\log_b x = y\) in logarithmic form.
Here is how you can convert from one form to another:

• Identify the base \(b\), the exponent or power \(y\), and the result \(x\) from the exponential equation.
• Rearrange these elements to match the logarithmic format. For example, an exponential equation \(e^{-0.9} = 0.406\) is converted to \(\log_e 0.406 = -0.9\).
• Recognize that when the base is \(e\), you can use the natural logarithm notation, thus \(\ln 0.406 = -0.9\).

The ability to switch effortlessly between these forms not only aids in solving equations but also enhances understanding of the underlying mathematical concepts.