Problem 94 - Elementary and Intermediate Algebra

Question

a. \(5^{9 x-1}=125\) b. \(5^{9 x-1}=124\)

Step-by-Step Solution

Answer

a. \(x = \frac{4}{9}\); b. \(x \approx 0.556\).

Step 1: Identify the RHS as a Power of 5 for Part (a)

For the equation \(5^{9x-1} = 125\), recognize that 125 can be rewritten as a power of 5: \(125 = 5^3\).

Step 2: Set Exponents Equal for Part (a)

Since the bases are the same (both are 5), set the exponents equal to each other: \(9x - 1 = 3\).

Step 3: Solve for x in Part (a)

Add 1 to both sides of the equation to get \(9x = 4\), then divide each side by 9 to solve for \(x\): \(x = \frac{4}{9}\).

Step 4: Analyze the Equation for Part (b)

For the equation \(5^{9x-1} = 124\), note that 124 cannot be written as a power of 5, which means there is no whole number solution. Try estimating logarithmically if needed.

Step 5: Use Logarithms for Approximate Solution for Part (b)

Take \(\log\) of both sides of the equation to get \((9x-1)\log 5 = \log 124\).

Step 6: Solve Logarithmic Equation for Part (b)

Isolate \(9x - 1\) by dividing both sides by \(\log 5\): \(9x - 1 = \frac{\log 124}{\log 5}\).

Step 7: Solve for x in Part (b)

Add 1 to both sides and then divide by 9 to solve for \(x\): \(x = \frac{\log 124}{9\log 5} + \frac{1}{9}\). Use a calculator to approximate \(x \approx 0.556\).

Key Concepts

Solving Equations

When solving equations, the main goal is to find the value of the unknown variable that makes the equation true. Equations can be either simple, like linear equations, or more complex, like quadratic or exponential equations. To solve them:

• Start by simplifying both sides of the equation if necessary.
• Look for ways to get the variable by itself on one side of the equation.
• Apply inverse operations to isolate the variable. For example, use addition to counter subtraction or division to counter multiplication.

In exponential equations, such as in the given exercise, isolating the exponent first often requires recognizing the right-hand side as a power of the base. If both sides of the equation have the same base, the exponents can be set equal. This is called the "property of equality for exponential equations." Changes in complexity, like those seen in part (b) of the exercise, require more advanced tools like logarithms when exact solutions aren't possible.

Logarithms

Logarithms are incredibly useful in solving equations that are not easily simplified through other means. Logarithms are essentially the inverse operation of exponentiation. They answer the question, "To what power is the base raised to obtain a particular number?" Here’s a handy guide to understand them better:

• If you have an equation where the variable is in the exponent, taking the logarithm of both sides can help bring the variable down from the exponent. This is what was done in part (b) of the exercise.
• Using logarithms can transform the multiplication of numbers into addition, simplifying the equation considerably.
• When using logarithms, any common base is acceptable, but base 10 (common logarithm) and base e (natural logarithm) are most frequently used.

For instance, in part (b), the logarithm of both sides was taken to solve for the exponent involving the variable. This technique provides a solution even when the other side is not a perfect power of the base, showing the flexibility of logarithms in finding approximate solutions.

Powers and Exponents

Understanding powers and exponents is pivotal when working with exponential equations. They depict repeated multiplication of a number, known as the base. Let's explore:

• Exponents indicate how many times the base is multiplied by itself, e.g., in the expression \(5^3\), the base 5 is used as a factor three times.
• Powers make expressions of large products more manageable and concise.
• When the base is the same, you can set the exponents equal if you have an equation, as shown in part (a) of our exercise.

This principle is handy because it allows simplification and direct comparison just by focusing on the exponents. Powers and exponents are foundational to mastering more advanced topics like exponential growth, decay, and logarithmic equations. They give us powerful tools for modeling and solving real-world problems efficiently.