Problem 48 - Algebra and Trigonometry

Question

Solve each exponential equation . Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 2^{2 x}+2^{x}-12=0 $$

Step-by-Step Solution

Answer

Therefore, the solution to the equation is \(x = log_{2}(3)\), and there is one non-real solution. The Decimal approximation of \(log_{2}(3)\) to two decimal places is 1.59

Step 1: Conversion into Quadratic equation

Let's substitute \(2^{x}\) by \(y\) to get the quadratic equation in \(y\). Hence, with this substitution, the equation \(2^{2x}+2^{x}-12=0\) becomes \(y^{2}+y-12=0\)

Step 2: Solving the Quadratic Equation

Factor the quadratic equation to get \(y - 3\)(y + 4) = 0. So, the solutions are \(y = 3\) and \(y = -4\)

Step 3: Turn Back into Original Variable

Change the variable back from y to \(2^{x}\). This gives the two equations \(2^{x} = 3\) and \(2^{x} = -4\)

Step 4: Solving for x

Apply logarithms on both sides of \(2^{x} = 3\) to get \(x = log_{2}(3)\). Similarly, for \(2^{x} = -4\), as the base of a logarithm (2, in this case) cannot be a negative number, therefore this equation has no solution.

Step 5: Expressing Solution in Terms of log

The solution \(x = log_{2}(3)\) as desired is in common logarithm and there is no solution in natural logarithm

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental concept in algebra where the highest power of the variable is two. They are usually written in the standard form of \( ax^2 + bx + c = 0 \) where \( a \) is not equal to zero, and \( a \) , \( b \) , and \( c \) are constants.

When faced with an exponential equation that can be converted into a quadratic equation, it's like unwrapping a disguised math problem. By making a substitution - for example, setting \( y = 2^x \) - we transform the problem into a more familiar form. This clever algebraic trick allows us to apply well-known methods for solving quadratic equations, such as factoring or using the quadratic formula.

Factoring and Solving Quadratics

Once in the form of \( y^2 + by + c = 0 \), we can attempt to factor the quadratic equation as a first step to find the values of \( y \). By setting each factor equal to zero, we can solve for \( y \).

• If the equation factors nicely, we'll get quick and clear solutions.
• If not, we can use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), to find the roots.

After solving for \( y \), we then substitute back to find the solutions for \( x \). This gives us the power to tackle non-traditional problems by leveraging the familiarity of quadratic equations.

Logarithms Simplify Exponential Equations

Logarithms act as a translator between the exponential and linear realms. They answer the question: 'To what exponent do we need to raise a base number to get another number?' The logarithmic form \( \log_{b}(x) \), is equivalent to the exponential form \( b^y = x \), where \( b \) is the base, \( y \) is the exponent, and \( x \) is the result.

Applying Logarithms

In the context of our problem, once we have the equation \( 2^{x} = 3 \) isolated, taking the logarithm of both sides allows us to bring down the exponent, thus linearizing what was an exponential relationship. We use the property \( \log_{b}(b^x) = x \) to solve for \( x \).

• For \( 2^{x} = 3 \), applying \( \log_{2} \) of both sides gives us \( x = \log_{2}(3) \).
• A common logarithm or natural logarithm can be used if the base \( b \) of the logarithm matches the base of the exponent.
• If the equation results in an impossible scenario like \( 2^{x} = -4 \) (since bases are positive in exponential functions), we can conclude there's no real solution.

Understanding the use of logarithms is key to simplifying and solving complex exponential equations efficiently.

Exponential Function Paves Way to Growth and Decay

Exponential functions, denoted as \( b^x \), are powerful tools used to describe a wide range of phenomena, including population growth, radioactive decay, and interest compounding. The base \( b \) of the exponential function is a constant, and \( x \) is the exponent.

Characteristics of Exponential Functions

These functions grow or decay at rates proportional to their current value, which makes them unique from linear functions. Their graphs show a distinctive 'J' shape, either increasing rapidly or decreasing towards zero without ever reaching it.

• When the base \( b > 1 \), the function models growth.
• When \( 0 < b < 1 \), it represents decay.

Having a solid grasp of exponential functions is crucial for solving exponential equations. When faced with an equation like \( 2^{2x} + 2^x - 12 = 0 \) from the exercise, we can identify the properties of the exponential function and apply algebraic techniques to solve for the variable \( x \).

By understanding their behavior, students are also better prepared to analyze real-world situations that can be modeled by exponential functions, offering a pathway to practical applications of mathematics beyond the classroom.