Problem 5 - Thomas Calculus

Question

In Exercises \(5-10,\) solve for \(y\) in terms of \(t\) or \(x,\) as appropriate. $$ \ln y=2 t+4 $$

Step-by-Step Solution

Answer

The solution is \(y = e^{2t + 4}\).

Step 1: Identify the Equation Type

The given equation is \(\ln y = 2t + 4\). It involves the natural logarithm, \(\ln\), which suggests that we need to solve for \(y\) by using the property of logarithms.

Step 2: Use Exponential Form

Recall that if \(\ln a = b\), then \(a = e^b\). Apply this property to convert the equation \(\ln y = 2t + 4\) into its exponential form. This gives us: \[ y = e^{2t + 4} \].

Step 3: Simplify the Exponential Expression

The equation \( y = e^{2t + 4} \) is already in its simplest form. This represents \(y\) in terms of \(t\) as required.

Key Concepts

Natural Logarithm

Natural logarithms are a critical concept in mathematics, often abbreviated as \( \ln \). Their base is the irrational number \( e \), approximately equal to 2.718. Natural logarithms are a tool to simplify calculations involving exponential growth or decay because they transform multiplicative processes into additive ones.
Natural logarithms are frequently used in solving equations where the unknown is in a power, such as in the equation \( \ln y = 2t + 4 \). Here, the natural logarithm helps isolate \( y \) by allowing us to use its exponential property to simplify the expression.
To use natural logarithms effectively, one must understand their primary property: if \( \ln a = b \), then \( a = e^b \). This transformation is essential when solving equations involving logarithms and exponents since it enables the expression of \( y \) in terms of \( t \) or other variables.

Solving for Variables

Solving for variables is an important skill in algebra, allowing us to express one variable in terms of others. It involves manipulating an equation to isolate the desired variable on one side.
In the equation \( \ln y = 2t + 4 \), we aimed to solve for \( y \) in terms of \( t \). Initially, \( y \) is entangled in a logarithmic expression which can seem complex, but we utilize the properties of natural logarithms to transform it.
By applying the property \( \ln a = b \) implies \( a = e^b \), we managed to isolate \( y \), transforming our original equation into an exponential form: \( y = e^{2t + 4} \). This move effectively expresses \( y \) with respect to \( t \), demonstrating how formula manipulation and understanding of exponentials make it possible to extract variables in equations.

Exponential Functions

Exponential functions are expressed in the form \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm, a fundamental constant in mathematics. These functions model a vast array of phenomena, including population growth, interest compounding, and radioactive decay.
When transitioning from a natural logarithmic form to an exponential expression, like \( y = e^{2t + 4} \), it's essential to grasp how exponentials function. They grow rapidly, reflecting very different kinetic properties than linear or polynomial equations. In this conversion, the parameter \( 2t + 4 \) determines the rate and starting point of the function with respect to \( t \).
Exponential equations are powerful in expressing relationships where change occurs continuously and multiplicatively. Solving them requires diving into the essence of exponents and the natural exponential base \( e \), a cornerstone in various mathematical and real-world applications.