Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation for \(x\). $$10^{2 x-1}=10^{x+3}$$

Step-by-Step Solution

Answer

The short answer to the equation \(10^{2x - 1} = 10^{x + 3}\) is \(x = 4\), by setting the exponents equal to each other and solving for 'x'.

Step 1: Identify the equal base

The equation has the same base (10) on both sides: \(10^{2x - 1} = 10^{x + 3}\)

Step 2: Apply the rule

Since the bases are equal, we´ll set the exponents equal to each other: \(2x - 1 = x + 3\)

Step3: Isolate 'x'

Now, we need to isolate 'x' and solve for it. First, let's subtract 'x' from both sides: \((2x - x) - 1 = x - x + 3\) \(x - 1 = 3\)

Step 4: Add 1 to both sides

To isolate 'x', we need to add 1 to both sides of the equation: \(x - 1 + 1 = 3 + 1\) \(x = 4\)

Step 5: Solution

The solution of the equation \(10^{2x - 1} = 10^{x + 3}\) is \(x = 4\).

Key Concepts

Exponential Function Properties

Exponential functions, such as the one involved in the exercise, have unique characteristics that make them distinct and interesting. One of the core properties is their base; for instance, the equation given here has a base of 10.

The function is written as: \(a^x\), where \(a\) is the base and cannot be 1 or a negative number, and \(x\) is the exponent, which can be any real number. The property that is often used to solve exponential equations is that if we have an equation where the exponential expressions have the same base and are set equal to each other, like \(10^{2x-1} = 10^{x+3}\), we can then deduce that their exponents must also be equal.

This property leads us directly into understanding how isolating variables is a strategic step in finding solutions to such equations.

Isolating Variables in Equations

To solve an equation for a specific variable means to isolate that variable on one side of the equation. This process often involves a series of algebraic steps. The key is to perform the same operation on both sides of the equation to maintain equality.

As shown in the step-by-step example, to solve for \(x\), you subtract \(x\) from both sides to combine like terms and then add or subtract other numbers to completely isolate \(x\). In our case, after subtracting \(x\) from both sides and adding 1, \(x\) stands alone.

\textbf{Useful tips for isolation}:

• Remember to do the same to both sides to keep the equation balanced.
• Use inverse operations: addition against subtraction, multiplication against division.
• Keep track of positive and negative signs to avoid errors.
• Check your solution by plugging it back into the original equation.

By mastering this technique, you'll be able to solve not only exponential equations but a wide variety of algebraic equations.

Logarithmic Equations

Logarithmic equations are closely related to exponential equations; they’re essentially the inverse. While the exponential equation we solved looks like \(10^{y}=x\), a logarithmic equation would be written as \(y=\log_{10}(x)\). It’s saying that \(10\) raised to what power equals \(x\)?

Solving logarithmic equations involves using logarithms to isolate the variable of interest, particularly when the variable is the exponent in an exponential function. There are properties of logarithms, such as the product, quotient, and power rules, that can simplify complex logarithmic expressions, but the core principle in solving them is similar to solving exponential equations—transform the equation to isolate the variable.

When facing more complex exponential equations, the method of taking the logarithm of both sides is useful, because it utilizes the property that \(\log_{b}(b^{x})=x\). This can help transform the equation into a form where the variable can be isolated and solved.