Problem 19 - Algebra A Combined Function

Question

Write each as a logarithmic equation. $$ 10^{2}=100 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the base

In the expression given, identify the base of the exponent. For this exercise, the base is 10.

Step 2: Identify the exponent

Determine the exponent in the equation, which is the power to which the base is raised. Here, the exponent is 2.

Step 3: Identify the result

Find the result of the exponential equation, which is what the base raised to the power equals. Here, the result is 100.

Step 4: Write the logarithmic equation

The formulation for a logarithmic equation is \( \log_{\text{base}}(\text{result}) = \text{exponent} \). Substituting the values from the previous steps, we get \( \log_{10}(100) = 2 \).

Key Concepts

Exponential Equations

Exponential equations are expressions where numbers are defined using a base raised to an exponent, written in the form \( a^b = c \). In our original exercise, the exponential equation is \( 10^2 = 100 \). This means that the base, 10, is multiplied by itself two times to get the result, 100.

Exponential equations are crucial because they help us represent large numbers more compactly, which is very useful in fields like science and engineering. They demonstrate how repeated multiplication can succinctly describe relationships between numbers.

Understanding exponential equations allows you to transform these expressions into other forms, such as logarithmic equations, which offer a different perspective on the relationship between numbers.

Bases and Exponents

The base in an exponential equation is the number that is repeatedly multiplied. In the equation \( 10^2 \), the base is 10.

The exponent indicates the number of times the base is used as a factor. It sits neatly as a small number above and to the right of the base. In the exercise example \( 10^2 \), the exponent is 2. This means that \( 10 \) is used twice in a multiplication, yielding \( 10 \times 10 = 100 \).

• The base provides the foundation for any exponential expression.
• The exponent modifies how many times the base is multiplied by itself.

Understanding bases and exponents helps in navigating between exponential and logarithmic forms, as it forms the primary connection in converting one to the other.

Logarithms

Logarithms are mathematical operations that help us find the exponent that a certain base is raised to, in order to obtain a specific number. They offer an alternative way to look at the relationships described by exponential equations. The core idea is that if \( a^b = c \), then \( \log_a(c) = b \).

In the example \( 10^2 = 100 \), the logarithmic form of this expression is \( \log_{10}(100) = 2 \). This states that 10 must be raised to the power of 2 to get 100.

• Logarithms are the inverse operations of exponents.
• They simplify the multiplication of large numbers into addition.
• This transformation helps in solving exponential equations in a different context.

By using logarithms, you simplify the process of solving exponential expressions and can gain better insights into complex calculations.