Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in logarithmic form. \(e^{k}=h\)

Step-by-Step Solution

Answer

\(\ln(h) = k\)

Step 1: Identify the Exponential form

The given equation is in the form of an exponential equation, written as \(e^k = h\). Here, \(e\) is the base of the exponential equation, \(k\) is the exponent, and \(h\) is the result or product of the exponential expression.

Step 2: Apply the Definition of Logarithms

To rewrite the equation in logarithmic form, use the definition of a logarithm which states that if \(b^x = y\), then the logarithmic form is \(\log_b(y) = x\). This allows us to convert the exponential equation into its logarithmic equivalent.

Step 3: Rewrite the Equation in Logarithmic Form

Using the logarithmic definition from the previous step, identify the parts: base \(e\), result \(h\), and exponent \(k\). Therefore, the logarithmic form of \(e^k = h\) is \(\ln(h) = k\), where \(\ln\) represents the natural logarithm with base \(e\).

Key Concepts

Exponential Form

Let's start by understanding the exponential form. An equation in exponential form looks like this:

• Base^Exponent = Result
• In symbols: \( b^x = y \)

A common example is \( 2^3 = 8 \), where 2 is the base, 3 is the exponent, and 8 is the result.
In the context of our exercise, the exponential form is \( e^k = h \), where \( e \) is a special number approximately equal to 2.718, often used in natural logarithms.
The exponential form is crucial because it lays the groundwork for converting into logarithmic form. This understanding helps us to transition smoothly between these forms.

Natural Logarithm

The natural logarithm, symbolized as \( \ln \), is a specific type of logarithm that uses \( e \) as its base.

• \( \ln(y) \) is equivalent to asking: "To what power must \( e \) be raised, to yield \( y \)?"
• For our equation \( e^k = h \), the natural logarithm provides: \( \ln(h) = k \)

Natural logarithms are widespread in mathematical and scientific applications because the base \( e \) appears naturally in a variety of contexts such as growth processes and radioactive decay.
Understanding the concept of a natural logarithm is essential when dealing with exponential equations involving \( e \). It helps simplify complicated exponential equations by converting them into manageable logarithmic forms.

Exponential Equation

An exponential equation includes an equation where a variable is in the exponent. The general form is \( b^x = y \).

• Exponential equations often appear in problems involving growth or decay, such as population growth or financial interest calculations.
• Our specific example \( e^k = h \) is a classic exponential equation.

Solving exponential equations often requires converting them into logarithmic form as it makes the exponents easier to handle.
These equations are quite powerful in mathematical modeling due to their ability to describe rapid changes in systems.

Logarithm Definition

The definition of a logarithm is fundamental to understanding their use and manipulation. For an equation \( b^x = y \), the logarithm can be expressed as \( \log_b(y) = x \). This means that \( x \) is the power to which the base \( b \) must be raised to result in \( y \).

• Logarithms operate as inverse operations to exponentials.
• In our exercise, using base \( e \), the conversion \( e^k = h \) becomes \( \ln(h) = k \).

This definition allows you to switch between exponential and logarithmic forms efficiently.
Once you understand the definition of a logarithm, solving and interpreting these problems becomes significantly simpler.