Problem 15 - Precalculus

Question

Write each logarithmic equation in its equivalent exponential form. $$\ln 5=x$$

Step-by-Step Solution

Answer

The equivalent exponential form is \( 5 = e^x \).

Step 1: Understand the Natural Logarithm

The natural logarithm, represented by \( \ln \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). When we see \( \ln 5 = x \), it means we are dealing with a logarithm of base \( e \).

Step 2: Express the Logarithmic Equation as \( y = \ln a \)

Given the equation \( \ln 5 = x \), we can express it in the general form \( x = \ln a \), where \( a = 5 \). This prepares us to rewrite the equation in exponential form.

Step 3: Convert to Exponential Form

Using the logarithmic to exponential conversion formula, \( y = \ln a \) can be written as \( a = e^y \). Therefore, the equation \( x = \ln 5 \) is equivalent to \( 5 = e^x \).

Step 4: Confirm the Exponential Equation

Revisiting the process, \( \ln 5 = x \) correctly converts to \( 5 = e^x \), meaning exponentiating \( e \) to the power \( x \) gives us \( 5 \). Hence, our exponential conversion is confirmed.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted by the symbols \( \ln \), is a logarithm with a special base known as \( e \). This base, \( e \), is an irrational constant approximately equal to 2.71828. Understanding the natural logarithm is essential because it is widely used in mathematics, particularly in calculus and mathematical modeling. It provides a way to describe exponential growth or decay, which is common in fields ranging from biology to economics.

• The natural logarithm of a number \( a \) answers the question: To what power should \( e \) be raised to obtain \( a \)?
• For example, in the equation \( \ln 5 = x \), what we mean is that \( e \) raised to the power of \( x \) will equal 5.

The reason the natural logarithm is so important is because \( e \) itself is one of the fundamental numbers in mathematics, much like \( \pi \). It frequently appears in problems involving continuous growth or compounding, making it an invaluable tool for understanding change over time.

Logarithmic Equation

A logarithmic equation involves a logarithm on one side of the equation. It is a vital concept because these equations can be solved to find unknown values in exponential relationships.

• Logarithmic equations allow us to rewrite and simplify expressions with exponents.
• In the context of the problem, \( \ln 5 = x \) is a simple logarithmic equation, with the logarithm base being \( e \).

By understanding logarithmic equations, one can transform them into a more familiar or simpler form, such as linear or exponential equations.The process typically involves identifying the base of the logarithm and then converting the equation into its exponential form. This conversion is often necessary to solve the equation for the unknown variable, making it a powerful tool in algebra and beyond.

Exponential Equation

An exponential equation features an unknown variable in the exponent, presenting a model for various growth and decay problems in mathematics and natural sciences.For instance, the result of our conversion from \( \ln 5 = x \) is the exponential equation \( 5 = e^x \).

• This means raising \( e \) (which is roughly 2.71828) to the power \( x \) gives us 5.
• Such equations are central in processes ranging from calculating interest to modeling population growth.

Understanding exponential equations allows students to grasp how a small change in the exponent can lead to significant results, especially as the value of \( e \) is raised to higher powers. This flexibility makes exponential equations crucial in scenarios dealing with rapid change or scaling effects.The transformation from logarithmic to exponential form is essential in solving equations and understanding relationships in data, thus supporting accurate predictions and analyses.