Problem 24 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \( x = \frac{\ln(15) - 3}{3} \).

Step 1: Isolate the Exponential Expression

Start by isolating the exponential part of the equation. Add 7 to both sides of the equation:\[4e^{3x+3} = 60\]

Step 2: Divide to Simplify

Divide both sides by 4 to solve for the exponential expression:\[e^{3x+3} = 15\]

Step 3: Apply the Natural Logarithm

Take the natural logarithm of both sides to eliminate the exponent. The natural logarithm, \( \ln \), and the exponential function \( e \) are inverse functions:\[\ln(e^{3x+3}) = \ln(15)\]

Step 4: Simplify the Logarithmic Expression

Apply the property of logarithms \( \ln(e^a) = a \). So,\[3x + 3 = \ln(15)\]

Step 5: Solve for x

First, subtract 3 from both sides:\[3x = \ln(15) - 3\]Then, divide by 3:\[x = \frac{\ln(15) - 3}{3}\]

Key Concepts

Exponential Equations

Exponential equations are equations where the variable is in the exponent. They often involve the base of the natural logarithm, which is denoted as \( e \). These types of equations can seem challenging, but with the right techniques, they become much simpler to solve.
Here are some tips for solving exponential equations:

• Isolate the exponential expression. Try to get the exponential part of the equation by itself on one side of the equation.
• Use logarithms to eliminate the exponent. Taking the logarithm of both sides allows you to bring down the exponent and work with it algebraically.
• Simplify and solve. After using logarithms, simplify the expression further to solve for the variable.

In the given problem, we first isolated \( e^{3x+3} \) as follows: \( e^{3x+3} = 15 \). This is a crucial step because it prepares the equation for further simplification using logarithms.

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a specific type of logarithm that uses the base \( e \). The number \( e \) is approximately 2.71828 and it's a fundamental constant in mathematics, much like \( \pi \). Natural logarithms are particularly useful in solving exponential equations involving the base \( e \).
Here's why natural logarithms are important:

• They are the inverse of exponential functions, specifically when the base is \( e \).
• Applying natural logarithms to both sides of an exponential equation allows you to simplify and solve for the variable in the exponent.

For example, in our exercise, after isolating the exponential expression, we took the natural logarithm of both sides: \( \ln(e^{3x+3}) = \ln(15) \). This step transforms the exponential equation into a more manageable form.

Inverse Functions

Inverse functions reverse the effect of a function. In the context of our problem, the natural logarithm (\( \ln \)) and the exponential function with base \( e \) are inverse functions. This means they undo each other.
Understanding inverse functions is key to solving certain types of equations:

• Exponential and logarithmic functions are inverses. If you take the logarithm of an exponential, you essentially "cancel out" the exponent.
• Property usage: When you apply the property \( \ln(e^a) = a \), it simplifies the equation significantly, allowing you to isolate the variable.

In the step \( \ln(e^{3x+3}) = 3x+3 \), we utilized this inverse relationship. It allowed us to eliminate the base \( e \) and directly solve for the variable \( x \), showing how powerful the concept of inverse functions can be in solving mathematical problems.