Problem 532 - Precalculus

Question

47\. Use the definition of a logarithm to find the exact solution for \(9+6 \ln (a+3)=33\).

Step-by-Step Solution

Answer

The exact solution is \( a = e^4 - 3 \).

Step 1: Isolate the Logarithmic Term

First, subtract 9 from both sides of the equation to isolate the logarithmic term. The equation becomes: \[ 6 \ln(a+3) = 24 \]

Step 2: Divide to Solve for \\( \ln(a+3) \\)

Divide both sides of the equation by 6 to solve for \( \ln(a+3) \). The equation becomes: \[ \ln(a+3) = 4 \]

Step 3: Rewrite the Equation in Exponential Form

Use the definition of a logarithm: \( \ln(a+3) = 4 \) is equivalent to \( a+3 = e^4 \). Thus, convert the logarithmic form to exponential form: \[ a + 3 = e^4 \]

Step 4: Solve for \\( a \\)

Subtract 3 from both sides to solve for \( a \):\[ a = e^4 - 3 \]

Key Concepts

Isolating the Logarithmic Term

Logarithmic equations can look a bit intimidating, but breaking them down step by step makes them manageable. The first step in solving a logarithmic equation, like the given problem, is to isolate the logarithmic term. This means we need to get the logarithmic expression by itself on one side of the equation. To do this for the equation \(9 + 6 \ln(a + 3) = 33\), we start by subtracting 9 from both sides. This gives us:

• \(6 \ln(a + 3) = 24\)

This step is crucial because it simplifies the problem, focusing our attention on the logarithmic part.
Next, divide both sides by 6, which simplifies to:

• \(\ln(a + 3) = 4\)

This method clears up the equation by focusing solely on the log expression, making the subsequent steps smoother.

Converting to Exponential Form

Once the logarithmic expression is isolated, the next move is to convert it into an exponential form, because this form can be easier to solve. Remember, the natural logarithm, represented by \(\ln\), has a base of \(e\), the mathematical constant approximately equal to 2.71828. In our equation \(\ln(a + 3) = 4\), the definition of logarithms tells us this is equivalent to:

• \(a + 3 = e^4\)

This step transforms the equation from logarithmic to exponential, helping us see the relationship between the terms more clearly. Converting to exponential form is like unwrapping the problem, providing a simpler, linear equation which is often easier to manage and solve.

Solving for the Variable

Now that the equation is in exponential form \(a + 3 = e^4\), our task is to solve for the variable \(a\). All we need is simple arithmetic to isolate \(a\). We do this by subtracting 3 from both sides, leading to:

• \(a = e^4 - 3\)

Remember, \(e^4\) is a number (approximately 54.598), so \(e^4 - 3\) can be calculated as well.
This step wraps up the solution by finding the exact value of \(a\). When you've worked through isolating the logarithmic term and transforming it into an exponential equation, the final step is often this type of straightforward arithmetic. Solving for the variable completes the journey from a potentially complex logarithmic problem to a clear, numerical answer.