Problem 58 - Algebra 2

Question

Solve each equation. $$ \ln (5 x-3)^{\frac{1}{3}}=2 $$

Step-by-Step Solution

Answer

The solution to the given equation is \( x \approx 2.078 \)

Step 1: Transform from Logarithmic to Exponential Form

The equation \( \ln (5x - 3)^{\frac{1}{3}} = 2 \) is in natural logarithmic form. This can be transformed into exponential form as following: \( e^2 = 5x - 3 \). Here, the base 'e' is raised to the power of 2, which equals the argument of the ln function.

Step 2: Evaluate e^2

Calculate the exponential \( e^2 \), approximately 7.3890561.

Step 3: Remove the Cube Root

By cubing both sides of the equation \( (7.3890561)^{\frac{1}{3}} = 5x - 3 \), we eliminate the cube root on left side. Then our equation will become: \( 7.3890561 = (5x - 3) \).

Step 4: Solve for x

Move -3 from the right side of the equation to left: \( 7.3890561 + 3 = 5x \). Then, divide by 5 on both sides to solve for x: \( x = \frac{10.3890561}{5} \)

Key Concepts

Natural Logarithms

In mathematics, logarithms help us understand and work with exponential relationships in a different way. The natural logarithm, denoted as \( \ln \), is a special logarithm with the base \( e \). The number \( e \) is approximately equal to 2.71828 and is an irrational number. Natural logarithms are incredibly important in various fields like calculus, economics, and even biology.

• Natural logarithms transform multiplication into addition, making it easier to solve complex problems.
• When you see \( \ln(x) \), it's asking: "To what power must \( e \) be raised to result in \( x \)?"
• This is essential for solving equations where variables appear in exponents, as logarithms can simplify these equations significantly.

Understanding natural logarithms is key to solving equations that might initially seem complicated, like the one in our exercise. In this case, converting \( \ln((5x - 3)^{\frac{1}{3}}) = 2 \) to exponential form helps us progress towards the solution by letting us see the underlying exponential relationship.

Cube Roots

Cube roots arise when you need to find a number that, when multiplied by itself three times, results in the original number. In mathematical notation, the cube root of a number \( a \) is written as \( a^{\frac{1}{3}} \). Cube roots are particularly useful when dealing with volumes and higher-dimensional problems.

• For instance, the cube root of 8 is 2, since \( 2 \times 2 \times 2 = 8 \).
• They provide a way to simplify algebraic expressions, especially in polynomial equations.
• In solving equations, knowing that cubing can "undo" a cube root allows you to isolate components step by step.

In the exercise, to eliminate the cube root on the left hand side of the equation, we apply the operation of cubing both sides. This allows us to transform \( (5x - 3)^{\frac{1}{3}} \) into \( 5x - 3 \), simplifying the path to the final solution of the equation.

Solving Equations

The art of solving equations lies in understanding and applying the rules of algebra to find the unknown variable. The process often involves isolating the variable on one side of the equation. Let's break it down with an example: first, understand the structure of the equation; next, simplify complex parts; and finally, isolate the variable to solve.

• Combine like terms and simplify complex expressions using logarithmic or algebraic transformations.
• For linear equations like \( ax + b = 0 \), rearrange terms to isolate \( x \).
• Ensure each operation is balanced by performing it on both sides.

In the exercise, these principles help turn \( 5x - 3 = e^2 \) into a solvable form. We adjusted the equation to get \( 5x = 10.3890561 \) by moving constants and dividing by coefficients, leading to the discovery of \( x \). Understanding each step in this process clarifies how different parts interact and how solutions come together.