Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, use logarithms to solve. \(4 e^{3 x+3}-7=53\)

Step-by-Step Solution

Answer

The value of \(x\) is approximately 0.255.

Step 1: Isolate the Exponential Term

Start by isolating the exponential term on one side of the equation. In the equation \(4 e^{3x+3} - 7 = 53\), first add 7 to both sides to get rid of the \(-7\). This gives us \(4 e^{3x+3} = 60\).

Step 2: Divide by the Coefficient

Next, divide both sides of the equation by 4 to further isolate \(e^{3x+3}\). This results in \(e^{3x+3} = 15\).

Step 3: Apply Natural Logarithm

To solve for \(3x+3\), apply the natural logarithm to both sides of the equation. We use the property \(\ln(e^y) = y\), which simplifies the left side: \(\ln(e^{3x+3}) = \ln(15)\) becomes \(3x+3 = \ln(15)\).

Step 4: Solve for x

Subtract 3 from both sides to solve for \(3x\), leading to \(3x = \ln(15) - 3\). Finally, divide both sides by 3 to find \(x\): \(x = \frac{\ln(15) - 3}{3}\).

Step 5: Compute the Numerical Value

Use a calculator to compute \(x\approx \frac{\ln(15) - 3}{3}\). Thus, the approximate value of \(x\) is calculated to a few decimal places.

Key Concepts

Exponential Equations

Exponential equations are equations where the variable appears in an exponent. These types of equations are common in various fields, such as biology, physics, and finance. In the equation \(4 e^{3x+3} - 7 = 53\), the variable \(x\) is in the exponent of base \(e\), which makes it an exponential equation. To solve exponential equations, you often need to perform operations to isolate the exponential component before applying logarithms. This step is crucial as it simplifies the complexity of handling variables within exponents. Additionally, understanding exponential equations is beneficial because they model growth and decay processes effectively, making them a strong tool in mathematical applications.

Natural Logarithm

The natural logarithm (denoted as \( \ln \)) is the logarithm to the base \(e\), where \(e \approx 2.718\). It is a critical function in solving equations involving the exponential constant \(e\). The property that \(\ln(e^y) = y\) makes it especially useful when dealing with exponential equations. In our example, after isolating the exponential term, we apply the natural logarithm: \(\ln(e^{3x+3}) = \ln(15)\). This step simplifies the equation to \(3x+3 = \ln(15)\), effectively removing the exponential component and making it linear. Thus, the natural logarithm acts as an essential tool for 'undoing' the exponential function, providing a pathway to solve for the variable.

Solving Equations

Equations are solved by manipulating them to find the value of the unknown variable. In our exponential equation, the solution process involved several key steps:

• Isolating the exponential term by adding and dividing.
• Applying the natural logarithm to both sides.
• Rearranging the equation to solve for \(x\).

Each step builds on the previous one, progressively simplifying the equation to reveal the solution. This systematic approach not only aids in finding the solution but also ensures accuracy. By following each step methodically and carefully, anyone can solve similar equations by applying these standard techniques.

Isolation of Terms

Isolation of terms is a foundational step in solving equations. It involves rearranging the equation so that the term containing the variable is on one side. In our problem, we first added 7 to both sides and then divided by 4, to isolate \(e^{3x+3}\). This process of isolation simplifies the equation, making it easier to apply subsequent operations such as logarithms. Proper isolation is key, as it sets up the equation to use specific algebraic operations or identities effectively. By understanding how to isolate terms, you can tackle and solve equations more efficiently, reducing errors and gaining a clearer grasp of mathematical relationships.