Problem 12 - A Graphical Approach to College Algebra

Question

For each statement, write an equivalent statement in exponential form. $$\log _{5} 5=1$$

Step-by-Step Solution

Answer

\( 5^1 = 5 \)

Step 1: Understanding the Logarithmic Form

The statement is given in logarithmic form as \( \log_{5} 5 = 1 \). In logarithms, \( \log_b a = c \) means that the base \( b \) raised to the power of \( c \) equals \( a \). Hence, \( 5 \) is the base, \( 5 \) is the argument, and \( 1 \) is the logarithm.

Step 2: Converting to Exponential Form

Using the definition of the logarithmic form, \( \log_{b} a = c \) is equivalent to saying \( b^c = a \). Thus, the given statement \( \log_{5} 5 = 1 \) is equivalent to saying \( 5^1 = 5 \).

Step 3: Checking the Exponential Form

Verify that the base \( 5 \) raised to the power of \( 1 \) indeed equals the result, \( 5 \). In this case, \( 5^1 = 5 \) holds true, confirming the correctness.

Key Concepts

Logarithmic Form

The logarithmic form expresses an equation where a base is raised to a certain power to produce a given number. In our example, we have the equation \( \log_{5} 5 = 1 \). This particular equation is saying that 5 is the base, and when raised to the power of 1, it equals 5. It's written as \( \log_{b} a = c \), where:\

• \( b \) is the base,
• \( a \) is the argument,
• \( c \) is the result of the logarithmic operation.

This is helpful to understand because it allows us to explore equations by moving between logarithmic and exponential forms.

Base and Argument

In the context of our logarithmic statement \( \log_{5} 5 = 1 \), the base and the argument are crucial components. The base \( b \) is the number we are using as the foundation for our power comparison—in this example, it's 5. The argument \( a \) is what the base raised to a power \( c \) equals—in our example, this is also 5. Here's what they signify:

• Base (\( b \)): The number that gets raised to the power.
• Argument (\( a \)): The result after the base is raised to the power.

Understanding the base and argument helps clarify how logarithms are used to solve equations and determine missing values. We use both logarithmic and exponential forms to switch between these when necessary.

Exponential Equations

Exponential equations frame our logarithmic expressions in a different light. They allow us to reinvestigate the relationship using powers. Given the statement \( \log_{5} 5 = 1 \), we convert it to its equivalent exponential form: \( 5^1 = 5 \). This conversion shows:

• The base \( 5 \) remains constant,
• The power \( 1 \) is applied to this base,
• The resulting value is 5, confirming the equality.

By checking the equation \( 5^1 = 5 \), we confirm that the exponential form is an accurate representation of the original logarithmic equation. This approach is vital for verifying solutions and understanding the underlying relationships between numbers.