Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(3\left(4^{x}\right)=81\)

Step-by-Step Solution

Answer

The solution to the equation is \(x=3\).

Step 1: Isolate the Exponential Term

Divide each side of the equation by 3, to isolate the exponential term: \(4^{x}=27\).

Step 2: Convert to the Same Base

To make it easier to handle, convert 27 into base 4, 27 is equal to \(3^{3}\) which is also equal to \((2^{2})^{3}\), or \(2^{6}\), So, the equation becomes \(4^{x}=2^{6}\).

Step 3: Solve for 'x'

Since \(4=2^{2}\), the left hand side of the equation can also be expressed as \((2^{2})^{x} = 2^{6}\). By the property of exponents, \(2^{2x} = 2^{6}\). Setting the exponents equal to each other gives: \(2x = 6\). Finally, solving this simple equation yields to: \(x=6/2\). which equal to 3.

Step 4: Check the Result

Puting x=3 into original equation \(3*(4^{3}) = 81\), which hold true, confirming that x=3 is a correct solution

Key Concepts

Algebraic Solutions

Solving exponential equations algebraically involves finding the unknown variable in an equation where the variable appears as an exponent. This process may seem complex initially, but breaking it down into simple steps can make it much more manageable. The solutions to exponential equations are often found by expressing both sides of the equation with the same base, allowing for the use of exponent rules to isolate and solve for the variable.

• Begin by isolating the exponential term, such as getting rid of any coefficients or terms on the opposite side of the equation.
• Express both sides of the equation with the same base.
• Utilize logarithmic functions if necessary to deal with bases that cannot be expressed this way.
• Directly compare or equate the exponents.

Finding these algebraic solutions requires a strong understanding of exponent rules and mathematical properties, which can transform seemingly complex problems into simpler forms.

Exponential Functions

Exponential functions are a cornerstone concept in algebra, characterized by their rapid rate of increase or decrease. An exponential function basically takes the form of \( a^{x} \), where \( a \) is a constant, and \( x \) is the variable exponent.

• Exponential growth occurs when the base, \( a \), is greater than 1.
• Conversely, exponential decay happens when \( a \) is between 0 and 1.

These functions are used to model a variety of real-world phenomena, such as population growth, radioactive decay, and financial interest formulas. Understanding how they work, including how to manipulate them algebraically, is crucial for solving any algebraic equations involving exponential terms. Another key feature of these functions is that their shape is not linear, but rather follows a rapid growth or decay, which makes them both fascinating and challenging to work with in mathematical contexts.

Equation Solving Steps

The process of solving exponential equations includes a few crucial steps that help ensure accuracy and understanding. The provided example of solving \( 3\left(4^{x}\right)=81 \) follows a structured approach which is typical for such problems.

• Step 1: Isolate the exponential term by performing operations that simplify the equation, such as dividing by a coefficient.
• Step 2: Rewrite both sides of the equation using the same base to leverage the properties of exponents. This step often requires expressing numbers as powers of smaller prime numbers.
• Step 3: Equate the exponents from the two sides once they share the same base, making it possible to solve for the unknown exponent.
• Step 4: Solve the resultant equation and verify your solution by substituting it back into the original equation to ensure it satisfies the condition.

These steps are helpful in systematically approaching exponential equations. With practice, recognizing when and how to apply these steps becomes more intuitive, allowing for quick and accurate solutions.