Problem 4 - Contemporary Precalculus

Question

Solve the equation without using logarithms. $$3^{7 x}=9^{2 x-5}$$

Step-by-Step Solution

Answer

Answer: The solution to the given equation is \(x = \frac{-10}{3}\).

Step 1: Express bases with a common base

As 3 is a factor of 9, we can rewrite the base of 9 as a power of 3. Since \(3^2 = 9\), we can rewrite the given equation as: $$3^{7x} = (3^2)^{2x - 5}$$

Step 2: Use exponent properties

By the property of exponentiation, we can multiply the exponents in the right side of the equation: $$3^{7x} = 3^{(2x - 5) \cdot 2}$$

Step 3: Equate exponents

Since the base is the same on both sides of the equation (3), we can conclude that the exponents must be equal: $$7x = (2x - 5) \cdot 2$$

Step 4: Expand and simplify

Now, we can expand and simplify the equation by solving for x: $$7x = 4x - 10$$ $$7x - 4x = -10$$ $$3x = -10$$

Step 5: Solve for x

Finally, divide by 3 to find the value of x: $$x = \frac{-10}{3}$$ Thus, the solution to the given equation is \(x = \frac{-10}{3}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a certain power, which is called the exponent. It is expressed as \( a^b \), where \( a \) is the base and \( b \) is the exponent. The operation represents repeated multiplication of the base. For example, \( 3^4 \) means multiplying 3 by itself four times, which equals 81.

• Base: The number being multiplied.
• Exponent: Indicates how many times the base is used as a factor.

Exponentiation is fundamental in solving equations where the variable is in the exponent, as growth and decay processes in natural sciences and financial calculations often follow exponential patterns.

Common Base

When solving exponential equations, converting different bases to a common base can significantly simplify the problem. In this exercise, we have the equation \( 3^{7x} = 9^{2x - 5} \). Notice that 9 can be expressed as a power of 3, specifically \( 9 = 3^2 \).

• This allows us to rewrite the equation: \( 3^{7x} = (3^2)^{2x - 5} \).
• Using a common base enables us to use properties of exponents to equate the exponents directly.

Converting to a common base is powerful because it allows comparison and simplification using algebraic rules rather than more complex logarithmic methods.

Equation Solving

Solving equations is about finding the value of the variable that makes the equation true. In exponential equations, especially when bases are matched, it can be simplified by equating the exponents.For this problem, after expressing 9 as a power of 3, we get \( 3^{7x} = 3^{4x - 10} \). Since the bases are the same (both are 3), the exponents on both sides must be equal.

• Equate the exponents: \( 7x = 4x - 10 \).
• Solve this simpler linear equation to find the solution for \( x \).

This reduces the complexity of the equation to that of solving a linear algebraic equation.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using basic algebraic rules to isolate the variable of interest. In our context, once the equation \( 7x = 4x - 10 \) is derived from equating exponents, the next steps involve manipulating this equation to solve for \( x \).

• Subtract \( 4x \) from both sides, resulting in \( 3x = -10 \).
• Divide both sides by 3, so \( x = \frac{-10}{3} \).

These manipulations are guided by the goal of isolating the variable, making it easier to determine its value. Algebraic manipulation is a versatile tool used across different types of mathematical problems.