Problem 55 - Algebra 2

Question

Solve each equation. $$ \ln x-3 \ln 3=3 $$

Step-by-Step Solution

Answer

The solution to the equation \( \ln x - 3 \ln 3 = 3 \) is \( x = 27e^3 \)

Step 1: Rewrite the Second Logarithm

The logarithm rule states that if \( \ln a^n = n \ln a \), so \( 3\ln3 \) can be rewritten as \( \ln3^3 \). Our equation becomes \( \ln x - \ln3^3 = 3 \)

Step 2: Merge the Logarithm using Logarithmic Laws

According to the logarithm rule \( \ln a - \ln b = \ln (a/b) \), we can rewrite our equation as \( \ln (x/3^3) = 3 \)

Step 3: Convert the Logarithm into Exponential Form

When converting from logarithmic to exponential form, the base of the log becomes the base of the exponent. The right-hand side of our equation is \( e^3 \). Our equation becomes \( x/3^3 = e^3 \)

Step 4: Solve for x

Multiply both sides by \( 3^3 \) or 27 to isolate \( x \) on the left-hand side. Our final answer then becomes \( x = 27e^3 \)

Key Concepts

Logarithmic Rules

To tackle logarithmic equations, understanding the basic rules of logarithms is crucial. These rules help simplify and manipulate logarithmic expressions, allowing us to work with them more easily. One of the key rules used in solving logarithmic equations is the power rule, which states:

• \( \ln a^n = n \ln a \)

This rule allows us to move coefficients in front of the \( \ln \) and convert them into exponents, aiding in simplification.
Another foundational rule is the quotient rule, which states:

• \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \)

This rule is beneficial for combining logarithms into a single term, making them easier to handle. By using these rules, you can transform complex logarithmic expressions into simpler forms, which is the first step when solving logarithmic equations like the one in our exercise.

Exponential Form

Once we have a simplified logarithmic equation, the next step is often to switch to exponential form. This transformation makes the equation easier to solve by removing logarithms. When converting a logarithmic equation to exponential form, remember that:

• If \( \ln a = b \), then you can write it as \( a = e^b \)

This conversion uses the natural base, \( e \), which is approximately 2.718. In the context of the exercise, after applying logarithmic rules, we reached the equation \( \ln \left( \frac{x}{3^3} \right) = 3 \). By converting this into exponential form, we get \( \frac{x}{27} = e^3 \), removing the logarithm and paving the way to solving for \( x \). This form helps in easily isolating the variable, which is the final goal.

Solving Equations

After converting the equation into exponential form, the equation becomes simpler and more straightforward to solve. Here's how it unfolds:
From \( \frac{x}{27} = e^3 \), our objective is to find \( x \). We achieve this by isolating \( x \) on one side of the equation:

• Multiply both sides by 27 to get \( x = 27e^3 \)

This step of multiplying both sides by the same number is a basic algebraic manipulation. It ensures that \( x \) is no longer divided by 27, thus solving our equation fully.
Understanding these steps in solving exponential equations is key as it involves simple algebraic techniques. Completing the exercise gives us \( x = 27e^3 \), which is your final answer. Following these methods consistently will aid in solving various logarithmic and exponential equations effectively.