Problem 17 - Calculus and its applications

Question

Write an equivalent logarithmic equation. $$ e^{t}=p $$

Step-by-Step Solution

Answer

The equivalent logarithmic equation is \(t = \ln(p)\).

Step 1: Identify the Exponential Form

First, recognize that the equation given is in the exponential form, which is expressed as \(e^{t} = p\). This form tells us that \(e\), which is the base of natural logarithms, is raised to the power of \(t\), resulting in \(p\).

Step 2: Rewrite using Logarithmic Form

To rewrite the exponential equation in logarithmic form, use the relationship between these forms: The exponential \(b^{x} = y\) can be rewritten in logarithmic form as \(x = \log_{b}(y)\). Specifically, for the given equation \(e^{t} = p\), it converts to \(t = \ln(p)\), where \(\ln\) denotes a natural logarithm with base \(e\).

Step 3: Confirm the Conversion

Check that the conversion aligns with logarithmic principles: The statement \(t = \ln(p)\) indicates the power you raise \(e\) to, in order to get \(p\). This validates our conversion from exponential to logarithmic form is correct and equivalent.

Key Concepts

Exponential Form

Understanding exponential form is crucial when dealing with equations involving exponents. In mathematics, an exponential equation typically looks like this: \( b^x = y \). In this format, \( b \) is the base, \( x \) is the exponent, and \( y \) is the result after \( b \) has been raised to the power of \( x \). In our exercise, the given equation \( e^t = p \) is in exponential form. Here, \( e \) is the constant base of natural logarithms, \( t \) is the exponent, and \( p \) is the result or outcome of this expression.
Exponential equations reveal the power relationships and how quantities grow rapidly, especially in contexts like population growth and compound interest. Understanding these equations helps in quickly switching to their corresponding logarithmic forms, aiding in solving various mathematical and real-world problems.

Natural Logarithms

Natural logarithms are a special type of logarithm where the base is \( e \), an irrational number approximately equal to 2.71828. The natural logarithm is commonly denoted as \( \ln(x) \) rather than \( \log_e(x) \), which makes it easier to read and write.
Natural logarithms have significant applications in mathematics, science, and engineering, particularly in calculus and growth processes.

• They make it straightforward to solve exponential equations with base \( e \), as seen in the exercise when converting \( e^t = p \) to \( t = \ln(p) \).
• They are often used for finding continuous growth or decay rates, which appear in many scientific fields like biology and finance.

By using natural logarithms, complex equations become more manageable, allowing for quicker and clearer solutions to problems using exponential equations.

Conversion between Forms

Converting between exponential and logarithmic forms is a valuable skill. It allows simplifying complex problems into more accessible forms. The basic principle revolves around the inverse nature of logarithms and exponents.
When you start with an equation in exponential form, such as \( b^x = y \), you can convert it to logarithmic form using \( x = \log_b(y) \). This conversion shows that the exponent \( x \) is equivalent to the logarithm of \( y \) with base \( b \).
In our specific case, converting \( e^t = p \) into its logarithmic equivalent \( t = \ln(p) \) demonstrates this process:

• Identify the base \( e \), exponent \( t \), and result \( p \).
• Translate this into the statement that \( t \) is the natural logarithm of \( p \).

This practice not only helps in reformatting equations to solve for unknowns but also enhances understanding of logarithmic algorithms and exponential functions.