Problem 20 - Calculus and its applications

Question

Write an equivalent logarithmic equation. $$ 10^{2}=100 $$

Step-by-Step Solution

Answer

\(\log_{10}(100) = 2\)

Step 1: Identify the components of the exponential equation

The given equation is in the exponential form: \(10^2 = 100\). In this form, 10 is the base, 2 is the exponent, and 100 is the result.

Step 2: Convert exponential form to logarithmic form

To convert the equation from exponential to logarithmic form, use the property that states: if \(a^b = c\), then \(\log_a(c) = b\). Here, \(a=10\), \(b=2\), and \(c=100\).

Step 3: Write the equivalent logarithmic equation

Using the property identified in Step 2, we convert the exponential equation \(10^2 = 100\) to the logarithmic form: \(\log_{10}(100) = 2\). This is the equivalent logarithmic equation.

Key Concepts

Exponential Form

In the realm of mathematics, the exponential form is a way to express numbers using a base and an exponent. When you see an expression like \(10^2\), you are looking at an exponential form equation. This means that the base, which is 10, is multiplied by itself 2 times. So, \(10^2\) equals 10 multiplied by 10, which gives us 100. This representation is compact, concise, and very powerful, especially when dealing with very large or very small numbers.

Understanding exponential form involves recognizing three main components:

• **Base:** The number that is being multiplied. In our example, it's 10.
• **Exponent:** The power to which the base is raised. Here, it's 2.
• **Result:** The outcome of the multiplication, which is 100.

Mastering this concept is crucial as it provides a foundation for moving onto logarithms and understanding how they function.

Exponential to Logarithmic Conversion

One of the fascinating aspects of mathematics is the ability to convert between exponential and logarithmic forms. This conversion helps us to view the same relationship from a different perspective. When you have an equation like \(10^2 = 100\) in exponential form, you can convert it to logarithmic form using the conversion rule: if \(a^b = c\), then \(\log_a(c) = b\).

Applying this rule here:

• The base \(a\) becomes the base of the logarithm.
• The result \(c\) is what you take the logarithm of.
• The exponent \(b\) is the result of the logarithmic operation.

This means that the exponential equation \(10^2 = 100\) can be expressed in logarithmic form as \(\log_{10}(100) = 2\). This conversion is valuable in solving equations where the unknown is the exponent.

Properties of Logarithms

Logarithms have unique properties that make them a powerful tool for solving equations involving exponents. These properties can simplify complicated expressions and aid in understanding exponential relationships. Here are some key properties:

• **Product Property:** \(\log_b(xy) = \log_b(x) + \log_b(y)\). This means that the logarithm of a product is the sum of the logarithms.
• **Quotient Property:** \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\). This indicates that the logarithm of a quotient is the difference of the logarithms.
• **Power Property:** \(\log_b(x^n) = n \cdot \log_b(x)\). This says that the logarithm of a power is the exponent times the logarithm of the base.

Understanding these properties allows us to manipulate and solve logarithmic expressions effectively. They serve as essential tools in expanding our ability to work with exponential and logarithmic equations.