Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln x+\ln x^{2}=3$$

Step-by-Step Solution

Answer

The solution is \(x = e\).

Step 1: Combine Logarithmic Terms

The equation is given as \(\ln x + \ln x^{2} = 3\). We can combine the logarithmic terms using the product rule for logarithms which states \(\ln a + \ln b = \ln(ab)\). Therefore, \(\ln x + \ln x^{2} = \ln(x \cdot x^{2}) = \ln x^{3}\). Now we have the simplified equation: \(\ln x^{3} = 3\).

Step 2: Convert to Exponential Form

To solve the logarithmic equation \(\ln x^{3} = 3\), we convert it to exponential form using the property that \(\ln a = b\) implies \(a = e^{b}\). Thus, \(x^{3} = e^{3}\).

Step 3: Solve for x

To isolate \(x\), take the cube root of both sides of the equation: \(x = (e^{3})^{1/3}\). Simplifying gives \(x = e\).

Step 4: Verification with Calculator

Using a calculator, verify the solution by computing \(\ln e + \ln e^{2}\). Since \(\ln e = 1\), the expression simplifies to \(1 + 2\cdot1 = 3\), which matches the original equation.

Key Concepts

Product Rule in Logarithms

In logarithmic equations, the product rule is a helpful property that allows us to combine logarithms with the same base. The rule simply states that the sum of two log terms can be converted into a single log term representing the product of their arguments. For instance, if we have

• \( \ln a + \ln b = \ln(ab) \).

This transformation is essential because it simplifies the logarithmic expression, making it easier to solve. In our original exercise, we encounter

• \( \ln x + \ln x^{2} \).

Applying the product rule turns this into a more manageable form:

• \( \ln(x \cdot x^{2}) = \ln x^{3} \).

This simplification brings us closer to solving the logarithmic equation. Understanding how to use the product rule can make solving these types of log problems much more straightforward and efficient. Whenever you see addition between logs, remember to check if the product rule can simplify your work!

Exponential Form Conversion

When solving logarithmic equations, converting them into exponential form is a crucial step. This conversion utilizes the relationship between logarithms and exponents. The rule indicates that an equation of the form

• \( \ln a = b \)

is equivalent to

• \( a = e^{b} \),

where \( e \) is the base of natural logarithms, approximately equal to 2.718. This step is fundamental in getting rid of the logarithm so that you end up with a more straightforward equation involving exponents that you can solve.
In our example,

• \( \ln x^{3} = 3 \),

converts to

• \( x^{3} = e^{3} \).

By removing the logarithmic function, we are free to manipulate the equation systematically using algebra, specifically solving for \( x \) by taking cube roots. With logarithmic equations, spotting where you can apply an exponential conversion can greatly simplify the problem at hand.

Verifying Solutions with Calculators

Checking the solution with a calculator is a great way to confirm that your answer is indeed correct. It involves plugging the solution back into the original equation and ensuring it satisfies that equation. Using the calculator's functions for natural logarithms can expedite this process. Here’s how you can verify: calculate each logarithmic part separately with your solution substituted in.
For example, in verifying

• \( x = e \),

we calculate

• \( \ln e + \ln e^{2} \).

Since we know that

• \( \ln e = 1 \),

this results in

• \( 1 + 2 \cdot 1 = 3 \).

If this matches the original equation's right side, we confirm the correctness of our solution. This step isn't just about confirming correctness; it's also about building confidence in approaching log problems. Trust your solution by verifying it—your calculator becomes a handy companion!