Problem 2 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ \ln 9=2.1972 \ldots $$

Step-by-Step Solution

Answer

The exponential form of the given logarithmic equation \(\ln 9 = 2.1972\) is \(e^{2.1972} = 9\)

Step 1: Identify the parts of the logarithmic equation

In the logarithmic equation \(\ln 9 = 2.1972\), 'e' is the base of the logarithm (since '\ln' implies a base of 'e'), 9 is the argument of the logarithm, and 2.1972 is the value of the logarithm.

Step 2: Convert to exponential form

Using the conversion rule from logarithmic form to exponential form (\(b^n = a\)), the given logarithmic equation \(\ln 9 = 2.1972\) can be converted into the exponential form \(e^{2.1972} = 9\). Note that 'e'=2.71828 approximately, which is the base of natural logarithms.

Key Concepts

Natural Logarithm

Understanding the concept of the natural logarithm is vital when dealing with logarithmic and exponential equations. The natural logarithm, represented as ln, is a specific type of logarithm with base 'e'. e is an irrational and transcendental number roughly equal to 2.71828. It occurs naturally in various scientific and mathematical contexts. For a positive number x, the natural logarithm ln(x) answers the question, 'To what power should we raise e to get x?'

For instance, if we say \( \ln(9) = 2.1972 \), we are stating that e raised to the power of approximately 2.1972 equates to 9. This relationship is foundational to converting between logarithmic and exponential forms, a key skill in understanding growth processes, compounding interest, and many natural phenomena where exponential changes occur.

Exponential Equations

Moving on to exponential equations, these involve variables in the exponent, as opposed to the base. The general form is \( b^x = a \), where b is the base, x is the exponent, and a is the result. To solve an exponential equation, we frequently need to express both sides of the equation with the same base or use logarithms to help isolate the variable.

Here’s a usual application: Given \( e^{2.1972} = 9 \), we're examining an equation where the base is 'e', and we're trying to find the power, 2.1972, which is the natural logarithm obtained from the inverse operation. Understanding exponential equations is crucial in fields as diverse as finance, biology, and physics, as they describe situations where something grows or decays at a rate proportional to its current value.

Logarithmic Equations

Finally, let's look into logarithmic equations, which have the general form \( \log_b(a) = x \). Here, we're looking for the power x to which we need to raise the base b to obtain the value a. Transferring between logarithmic and exponential forms is a common method used to solve these equations.

For example, the logarithmic equation \( \ln(9) = 2.1972 \) can be converted into an exponential equation to solve for the value previously known to be 9. This conversion is possible because the operations of logarithms and exponentiation are inverses of each other. Mastering logarithmic equations is essential for interpreting data that spans several orders of magnitude, such as pH levels in chemistry, the Richter scale for earthquakes, or decibel levels in sound intensity.