Problem 63 - Algebra 2

Question

Solve each equation. $$ \ln x+\ln 2=6 $$

Step-by-Step Solution

Answer

The solution to the given equation is \( x = e^6 / 2 \).

Step 1: Combine Logarithmic Terms

First, use the properties of logarithms to combine the terms on the left side of the equation. This gives \(\ln(2x) = 6 \).

Step 2: Convert To Exponential Form

Next, convert the equation from logarithmic form to exponential form. This gives \( e^6 = 2x \).

Step 3: Solve for x

Finally, rearrange the equation to solve for x. This gives \( x = e^6 / 2 \).

Key Concepts

Logarithmic Properties

Logarithmic properties are essential tools in simplifying complex logarithmic expressions. In our exercise, we come across the expression \( \ln x + \ln 2 \). The core property at play here is the **Product Rule of Logarithms**, which states that the sum of two logs with the same base can be combined.

Specifically, the formula is:

• \( \ln a + \ln b = \ln(ab) \)

This property helps in reducing multiple terms into a single logarithmic expression, making further manipulation much easier. Notice that we rewrote the expression as \( \ln(2x) \), combining both terms into a single log expression, preparing it for further operations.

Exponential Functions

Exponential functions are a crucial component when dealing with logarithmic equations. Once we have simplified the logarithmic expression using its properties, the next step is often to convert the equation into an exponential form. This aligns with the understanding that logarithms and exponentials are inverse operations.

In our specific problem, we have the equation \( \ln(2x) = 6 \). To transform this into its exponential counterpart, we recognize the logarithmic definition:

• If \( \ln a = b \), then \( a = e^b \).

Thus, \( \ln(2x) = 6 \) converts to \( 2x = e^6 \). This conversion is pivotal in making the equation solvable, transitioning from the logarithmic to the algebraic world.

Equation Solving Steps

Solving logarithmic equations requires a systematic approach, combining logarithmic properties and algebraic manipulation. Once your equation is transformed into exponential form, you can solve it using these clear steps:

Let's break down the steps:

• **Step 1:** Simplify the logarithmic terms using properties like the Product Rule.
• **Step 2:** Convert the logarithmic equation into exponential form, utilizing the relationship between logarithms and exponentials.
• **Step 3:** Isolate the variable of interest (in this case \( x \)).
• **Step 4:** Perform arithmetic operations as necessary to find \( x \), leading us in our exercise to solve \( 2x = e^6 \) and thus \( x = \frac{e^6}{2} \).

By following these steps diligently, solving logarithmic equations becomes a structured process, leading to clear and precise solutions.