Problem 257 - Calculus Volume 1

Question

For the following exercises, write the equation in equivalent logarithmic form. $$ 9^{0}=1 $$

Step-by-Step Solution

Answer

\(\log_9(1) = 0\)

Step 1: Identify the Exponential Form

The given equation is in exponential form: \(9^0 = 1\). Here, the base is 9, the exponent is 0, and the result is 1.

Step 2: Write the Logarithmic Form

To convert the exponential equation \(b^e = n\) to its equivalent logarithmic form, the equation is written as \(\log_b(n) = e\).

Step 3: Apply the Conversion

Using the rule from Step 2, substitute the values from the given problem: base \(b = 9\), exponent \(e = 0\), and the result \(n = 1\). The logarithmic form is \(\log_9(1) = 0\).

Key Concepts

Exponential Form

When we talk about the exponential form, we are referring to numbers expressed as a base raised to an exponent. It's a compact way to show repeated multiplication. For example, in the expression \(9^0\), "9" is the base and "0" is the exponent.
This means that the base, 9, is multiplied by itself zero times, which might sound a bit odd at first. However, remember that any number raised to the power of zero equals 1. Why? Because it's a fundamental property of exponents: no matter the base, any number with an exponent of zero is 1.
Exponential form:

• Simplifies writing very large or very small numbers
• Turns multiplication of the same number into an easier task
• Is widely used in science and engineering to express large powers

Base and Exponent

The base and exponent are key elements in understanding mathematical expressions involving powers. Let's break them down:
The **base** is the number that is being multiplied. In our earlier example, 9 is the base. It tells us the number that is repeatedly being multiplied. The **exponent** tells us how many times to multiply the base by itself. In \(9^0\), the exponent is 0. This means 9 is multiplied zero times, resulting in a value of 1.
Understanding base and exponent is crucial because:

• They form the foundation for other arithmetic operations like logarithms
• Together, they explain exponential growth and decay found in real-world scenarios
• Mastering them aids in learning more complex math concepts

Conversion of Equations

Converting an exponential equation into a logarithmic form is a common task in algebra. This is done to solve equations involving exponentials by finding unknown values of exponents.
To convert, recall the formula: if an exponential equation is written as \(b^e = n\), it turns into a logarithmic statement \(\log_b(n) = e\). In our example, \(9^0 = 1\) converts to \(\log_9(1) = 0\). Here, you simply transfer components: base remains base, result becomes the argument of the log, and the exponent goes to the other side of the equation.
The benefits of conversion are:

• Logarithmic form provides an alternative perspective to solve complex equations
• Simplifies calculations, especially when dealing with very large or small numbers
• Examples include computing time in exponentially growing scenarios, like interest rates