Problem 6 - Thomas Calculus

Question

In Exercises \(5-10,\) solve for \(y\) in terms of \(t\) or \(x,\) as appropriate. $$ \ln y=-t+5 $$

Step-by-Step Solution

Answer

\( y = e^{5-t} \).

Step 1: Understand the Given Equation

We are given the equation \( \ln y = -t + 5 \). The goal is to solve for \( y \) in terms of \( t \), which means we need to isolate \( y \) on one side of the equation.

Step 2: Apply Exponentiation to Remove the Natural Log

To eliminate the natural logarithm from the equation, we raise \( e \) (the base of the natural logarithm) to the power of both sides. This gives us \( y = e^{-t+5} \).

Step 3: Simplify the Expression

The expression \( e^{-t+5} \) can be rewritten using properties of exponents: \( y = e^{5-t} \). This is the expression for \( y \) in terms of \( t \).

Key Concepts

Solving for Variables

In algebra and calculus, solving for a variable means finding the value or expression for that variable in terms of other variables given in the equation. In our given problem, you are tasked to express \( y \) in terms of \( t \). The provided equation is \( \ln y = -t + 5 \). This is a natural logarithmic equation, and our goal is to isolate \( y \) to get an expression that tells us what \( y \) equals based on the value of \( t \).

• First, observe the equation \( \ln y = -t + 5 \), which indicates that \( \ln y \) (the natural logarithm of \( y \)) equals \(-t + 5\).


• To solve for \( y \), you need to "undo" the logarithm by using exponentiation. This involves raising the number \( e \) to both sides of the equation to isolate \( y \).

By doing this, you will transition from a logarithmic equation to an exponential one, which will directly give you the expression for \( y \) in terms of \( t \).

Natural Logarithm

The natural logarithm, denoted \( \ln \), is a logarithm with a base \( e \), where \( e \approx 2.71828 \). It is a mathematical tool used to help solve equations involving exponential growth or decay, among other applications.

• A natural logarithmic equation like \( \ln y = x \) means that \( y \) can be expressed as \( e \) raised to the power of \( x \). In other words, \( y = e^x \).


• Logarithms represent the power to which a base number must be raised to obtain a certain value. Here, the equation \( \ln y = -t + 5 \) means \( y \) is equal to \( e \) raised to the power \(-t + 5\).

Understanding natural logarithms is essential in solving equations where the rate of increase or decrease is proportional to the current value, common in scientific and financial calculations.

Exponentiation

Exponentiation is a mathematical operation that involves raising a base number to the power of an exponent. It is crucial for solving logarithmic equations such as the one we have here. In this context, exponentiation is used to "cancel out" the logarithmic function, effectively isolating the variable.

• For the equation \( \ln y = -t + 5 \), exponentiation by \( e \) is used to solve for \( y \): \( e^{\ln y} = e^{-t + 5} \).


• This transformation results in \( y = e^{-t + 5} \). You can further simplify this expression using the properties of exponents: \( y = e^{5-t} \).

Using exponentiation efficiently allows you to solve for variables that are inside a logarithm, making it a powerful tool in algebra and calculus. It's important to become familiar with exponent rules as they often simplify expressions and calculations.