Problem 25 - Elementary and Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ \log _{10} 10=1 $$

Step-by-Step Solution

Answer

The exponential form is \( 10^1 = 10 \).

Step 1: Understanding the Given Logarithmic Equation

The given equation is \( \log_{10} 10 = 1 \). This equation means that 10, the base of the logarithm, raised to the power of the result (which is 1), yields the argument of the logarithm, which is also 10.

Step 2: Identify Exponential Form Equation Pattern

The pattern for converting a logarithmic equation \( \log_b a = c \) to an exponential form is \( b^c = a \). In this pattern, \( b \) is the base of the logarithm, \( a \) is the argument, and \( c \) is the result of the logarithm.

Step 3: Apply the Pattern to Convert to Exponential Form

Using the pattern from Step 2, write the exponential form of the given logarithmic equation. Here, the base \( b = 10 \), the argument \( a = 10 \), and the result \( c = 1 \). The exponential form is \( 10^1 = 10 \).

Key Concepts

Logarithmic Form

The logarithmic form is a way of expressing a quantity in terms of a base raised to a power or exponent. It is a powerful tool, especially when dealing with very large or very small numbers. When you see a logarithmic equation like \( \log_{10} 10 = 1 \), it means "the power to which the base 10 must be raised to get 10 is 1".
This form emphasizes the relationship between a base and its exponent through a logarithm. It’s important because it allows complex exponential relationships to be expressed in a more manageable arithmetic form.

Key components of the logarithmic form include:

• Base (b): The number being raised to a power, like 10 in \( \log_{10} \).
• Argument (a): The number that results from raising the base to a certain power, such as 10 in the equation.
• Exponent (c): The result of the logarithm, showing the power to which the base must be raised to produce the argument, which is 1 in this example.

Understanding these parts individually will make it easier to convert between logarithmic and exponential forms.

Exponential Form

The exponential form of an equation is essentially the inverse of the logarithmic form. It expresses numbers as a base raised to an exponent. This form is often more intuitive for understanding the magnitude of numbers directly.
In the equation \( 10^1 = 10 \):

• Base (b): The number that is raised to the power, which is 10 in this example.
• Exponent (c): The power to which the base is raised, noted as 1.
• Value/Argument (a): The result of the exponential expression; in this case, it is also 10.

The beauty of the exponential form lies in its simplicity and ability to represent both large and small numbers effectively.

This form is critical in many areas like growth models in finance, population studies, and computing to solve compound interest problems. It closely aligns with how many naturally occurring systems grow or decay.

Conversion between Forms

Converting between logarithmic and exponential forms is a valuable skill in mathematics because it helps to translate a problem into the form that is easiest to solve.
Here is the general rule for conversion:

• From Logarithmic to Exponential: Convert \( \log_b a = c \) into \( b^c = a \).
• From Exponential to Logarithmic: Convert \( b^c = a \) into \( \log_b a = c \).

In our example of \( \log_{10} 10 = 1 \), you successfully translate this into the exponential form \( 10^1 = 10 \) by identifying that:

• b is 10.
• a is 10.
• c is 1.

This simple conversion reveals the essence of the relationship being expressed. It allows us to use exponential expressions to simplify multiplication and division problems.
Further, this skill of conversion is essential for algebraic manipulations and solving real-world problems that involve exponential growth or decay, making it a critical technique in both academics and various industries.