Problem 16 - Precalculus Functions and Graphs

Question

Determine the number that can be used in place of the question mark to make the equation true. $$\left(\frac{1}{5}\right)^{?}=\frac{1}{125}$$

Step-by-Step Solution

Answer

? = 3

Step 1 - Recognize the base

Identify the base of the expression on the left-hand side. The base in the left-hand side is \(\frac{1}{5}\).

Step 2 - Rewrite the right-hand side

Express the right-hand side of the equation in terms of the same base. \(\frac{1}{125}\) can be written as \(\left( \frac{1}{5} \right)^{3} \).

Step 3 - Equate the exponents

Since the bases are now the same, you can set the exponents equal to each other: \( ? = 3 \).

Key Concepts

Exponential Functions

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. It looks like this: \(a^x\). Here, 'a' is the base and 'x' is the exponent.
Exponential functions grow or shrink rapidly. They are used in many fields such as science, finance, and population modeling.
Examples of exponential functions include:

• Growth: \(2^x\)
• Decay: \(\frac{1}{3}^x\)

In the given exercise, the exponential function on the left-hand side is \((\frac{1}{5})^?\). This shows exponential decay because the base \(\frac{1}{5}\) is a fraction less than 1.

Base Recognition

Base recognition means identifying the base in an exponential function. Recognizing the base is the first step to solving exponential equations.
To solve \((\frac{1}{5})^? = \frac{1}{125}\), we need to know that the base on the left-hand side is \(\frac{1}{5}\).
Next, we must express the right-hand side in terms of the same base. Notice that \(125 = 5^3\). Therefore, \(\frac{1}{125} = (\frac{1}{5})^3\).
When both sides of the equation have the same base, it becomes easier to solve for the unknown exponent.

Equating Exponents

After rewriting both sides with the same base, you can equate the exponents. This is because the bases are already matched.
If we have \((\frac{1}{5})^x = (\frac{1}{5})^3\), then we can simply say that \(x = 3\).
This is known as 'equating exponents'. This step transforms the original exponential equation into a simple linear equation.
Here are the steps we followed:

• Recognize the base.
• Rewrite each side with the same base.
• Equate the exponents to solve for the variable.

This method is efficient and widely used for solving exponential equations.