Problem 7 - Precalculus with Limits

Question

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25 .\) \(\log _{4} 16=2\)

Step-by-Step Solution

Answer

The exponential form of the given equation \(\log _{4} 16=2\) is \(4^{2}=16\).

Step 1: Identify the base, argument, and result in the logarithmic equation

In the given equation \(\log _{4} 16=2\), the base of the logarithm is 4, the argument is 16, and the result (or answer) is 2.

Step 2: Apply the relationship between logarithms and exponentiation

We use the fact that if \(\log _{b} a=n \), then \(b^n=a\). In our case, \(b=4\), \(a=16\), and \(n=2\).

Step 3: Write in exponential form

Following the relationship from step 2, the exponential form of the given equation is \(4^{2}=16\).

Key Concepts

Logarithmic Equations

Logarithmic equations are equations involving logarithms. They let you write exponential relationships in a different way. For example, in the equation \(\log_4 16 = 2\), we have a log base of 4 and want to know which exponent (2 in this case) will make the base become 16.
To fully understand, know these terms:

• **Base:** The number being raised to a power, shown as 4 in \(\log_4 16 = 2\).
• **Argument:** This is what you get after applying the exponent to the base, which is 16 here.
• **Result (or Exponent):** The power that the base is raised to, which is 2 in this equation.

By solving logarithmic equations, you rewrite them in exponential form, revealing the power relationship between numbers.

Base and Exponent

In mathematics, understanding the base and exponent is crucial to manage both exponential and logarithmic equations effectively. A base is the number that gets multiplied, and an exponent tells how many times the base is used in multiplication.
Let's break down the key terms using the equation \(4^2=16\):

• **Base (4):** The starting number, which is repeatedly multiplied by itself.
• **Exponent (2):** Indicates how many times the base is used in the multiplication process.

The relation can be demonstrated as \(4 \times 4 = 16\). This simple multiplication shows why \(4^2\) equals 16. Remembering how to interpret base and exponent is essential for transitioning between logarithmic and exponential forms.

Logarithmic and Exponential Relationship

The relationship between logarithmic and exponential equations forms a fundamental concept in mathematics. Think of them as two sides of the same coin. They are ways to express the same relationship, just flipped.
Here's the general rule:

• **Logarithmic Form:** \(\log_b a = n\) means that base \(b\) raised to the power \(n\) gives \(a\).
• **Exponential Form:** \(b^n = a\) stems directly from the logarithmic form, simply expressing what the variable \(n\) does with base \(b\) to reach \(a\).

Understanding this interrelationship is key, as transitioning between forms often simplifies problem-solving. Practicing converting between these forms solidifies comprehension of how both expressions convey the same numerical truth.