Problem 46 - 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. $$ e^{4 x}-3 e^{2 x}-18=0 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x=ln(6)/2\) which equates to \(x=0.90\) when converted into a decimal approximation. There's no solution for \(e^{2x}= -3\) because the exponential function only returns positive values.

Step 1: Rewriting the Exponential Equation as a Quadratic Equation

The given equation is \(e^{4 x}-3 e^{2 x}-18=0\). Note, we can think of \(e^{2x}\) as the variable 'u'. Therefore making a substitution \(u= e^{2x}\), the equation is transformed into a standard quadratic equation \(u^2 -3u -18 = 0\).

Step 2: Solving the Quadratic Equation

This quadratic equation can be solved by factoring or using the quadratic formula. The equation factors to give \((u-6)(u+3)= 0\), which gives the solutions \(u=6\) and \(u=-3\).

Step 3: Un-substituting the Variable

We now replace back the \(u\) we introduced with \(e^{2x}\), to give \(e^{2x}= 6\) and \(e^{2x}= -3\).

Step 4: Applying the Logarithmic Function and Solving

We can now solve both equations for \(x\) by taking the natural logarithm on both sides. For \(e^{2x}= 6\), applying a natural logarithm on both sides we get \(2x = ln(6)\) which implies \(x= ln(6)/2\). For \(e^{2x}= -3\), since the exponential function only returns positive values, we have no solution.

Step 5: Calculating the Decimal Approximation

Through use of a calculator, the decimal approximation of \(x=ln(6)/2\), correct to two decimal places, gives \(x= 0.90\).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and have the general form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants. They are named quadratic because "quadratic" means "involving the second power." The highest exponent of the variable (usually \(x\)) is 2. Solving these equations often requires finding the values of \(x\) (the roots) that satisfy the equation.

There are several methods to solve quadratic equations:

• Factoring: This involves rewriting the equation in a product form, usually easier if the equation is simple or has integer roots.
• Quadratic Formula: Regardless of whether the equation is factorable, the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) always works. It directly computes the roots.
• Completing the Square: This method involves manipulating the equation into a perfect square trinomial, making it easy to take square roots and solve for \(x\).

These approaches are the building blocks for solving not just simple algebraic equations but also more complex ones that can be written in quadratic form, as in our original problem.

Natural Logarithms

Natural logarithms are logarithms with the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. This logarithm measures the time needed for a quantity to grow to its growth capacity where the growth rate is proportional to its current value.

The natural logarithm of a number \(x\) is written as \(ln(x)\). A natural logarithm function \(ln(x)\) is the inverse of the exponential function \(e^x\). For example, if \(y = e^x\), then \(x = ln(y)\).

Some key properties of natural logarithms include:

• Product Property: \(ln(ab) = ln(a) + ln(b)\)
• Quotient Property: \(ln\left(\frac{a}{b}\right) = ln(a) - ln(b)\)
  
• Power Property: \(ln(a^b) = b \cdot ln(a)\)

These properties make natural logarithms extremely useful in solving exponential equations, especially when isolating variables proved challenging using basic algebraic methods. They allow converting multiplicative and exponential relationships into additive and linear ones, which are typically easier to handle.

Factoring Equations

Factoring equations is a valuable technique for finding zeros or solutions of equations. It involves expressing an equation as a product of its factors. For instance, if we have an equation \(x^2 - 4 = 0\), it can be factored into \((x - 2)(x + 2) = 0\). Each factor set to zero gives the equation's solutions: \(x - 2 = 0\) resulting in \(x = 2\), and \(x + 2 = 0\) resulting in \(x = -2\).

Some essentials for factoring:

• Common Factoring: Removing common factors from terms to simplify an equation before further factoring.
• Difference of Squares: An equation like \(a^2 - b^2\) can be factored as \((a-b)(a+b)\).
• Trinomial Factoring: Involves breaking down an expression like \(ax^2 + bx + c\) into two binomials.

These techniques allow us to transform complex equations into simpler, more manageable factors. Once factored, solving these simpler equations is comparatively straightforward, as shown in our original problem. Factoring offers clarity, making it invaluable when tackling quadratic and polynomial equations.