Problem 47 - 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. $$ 3^{2 x}+3^{x}-2=0 $$

Step-by-Step Solution

Answer

The solution to the equation \(3^{2x} + 3^{x} - 2 = 0\) is \(x = 0\). No decimal approximation is needed as the solution is an integer.

Step 1: Simplification using substitution

To simplify the given equation, make a substitution. Let \(y = 3^{x}\). Substitute this into the given equation: \(y^2 + y - 2 = 0\)

Step 2: Solving the quadratic equation

Now, solve this quadratic equation \((y^2 + y - 2 = 0)\) for \(y\) using the factored form of quadratic equations: \((y - 1)(y + 2) = 0\). Then solve for \(y\): \(y=1\) or \(y=-2\).

Step 3: Convert back to terms of x

Substitute \(3^{x}\) back in for \(y\), to return the solution in terms of \(x\). So from \(y = 1\) we have \(3^{x} = 1\), thus \(x = 0\). And from \(y = -2\) we have \(3^{x} = -2\), however this cannot be solved since the exponential function is always positive. So, the only solution is for \(x = 0\).

Step 4: Decimal approximation

As \(x = 0\), the statement about the decimal approximation becomes redundant, because zero doesn't need any decimal approximation.

Key Concepts

Natural Logarithms

In dealing with exponential equations, the concept of logarithms becomes very important. A natural logarithm (denoted as \(\ln\)) is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. Natural logarithms are frequently used in mathematics to solve equations involving exponential growth or decay.

• When you have an equation of the form \(a^x = b\), you can solve for \(x\) by taking the natural logarithm of both sides: \(x = \ln(b) / \ln(a)\).
• This method is particularly useful when the base of the exponential function isn't easy to handle with simple arithmetic. By converting to logarithms, we can linearize exponential growth.

In the context of the given problem, solving for \(x\) once \(3^x = a\), where \(a\) is a known value, involves taking the natural logarithm.

Quadratic Equation

Quadratic equations are a type of polynomial equation of the form \(ax^2 + bx + c = 0\). These equations often arise in various geometry and algebra problems. Solving quadratic equations can often be achieved using factoring, completing the square, or the quadratic formula.

• In the step-by-step solution, the substitution \(y = 3^x\) transforms the original exponential equation into the quadratic equation \(y^2 + y - 2 = 0\).
• This specific quadratic equation can be factored into \((y - 1)(y + 2) = 0\). This means \(y = 1\) or \(y = -2\).
• Solving for \(y\) gives us potential solutions for \(3^x\), but because \(3^x\) is always positive, \(y = -2\) doesn't provide a valid solution.

Understanding how to manipulate and solve quadratic equations is crucial in breaking down complex higher order polynomials into manageable solutions.

Decimal Approximation

Once you have solved an equation symbolically, obtaining a decimal approximation can be useful for practical applications, such as calculating interest rates or population growth. Decimal approximations give us an easy-to-understand numeric result that doesn't require further interpretation.

• To obtain a decimal approximation, one generally uses a calculator after finding the principal solution symbolically. In typical cases, this means converting a logarithmic or irrational number result to a simple decimal.
• For the equation in the exercise, it's noted that the symbolic solution \(x = 0\) doesn't need a decimal approximation, as it already is a clear and concise integer.

In many rough calculations or where extreme precision is not necessary, a decimal approximation can provide a sufficient answer for most needs.

Exponential Function

Exponential functions are fundamental in mathematics, often featuring the famous base \(e\), or other constants like 2, 3, or more. These functions show how quantities grow or shrink quickly, making them crucial in calculus, physics, and finance.

• The general form of an exponential function is \(f(x) = a^x\), where \(a\) is a positive real number, and \(x\) is the exponent. This function is defined for all real \(x\) and gives rise to a curve that increases (or decreases) without bound as \(x\) moves.
• One of the most important characteristics of exponential functions is that they never yield negative values. This property was clear in the provided problem, where \(3^x = -2\) had no solution because you can't raise a positive base to any power to get a negative number.

Understanding how exponential functions behave allows students to better grasp growth models, sound waves, and even patterns in compound interest.