Problem 14 - Algebra for College Students

Question

Write each logarithmic statement in exponential form. For example, \(\log _{2} 8=3\) becomes \(2^{3}=8\) in exponential form. $$ \log _{5} 25=2 $$

Step-by-Step Solution

Answer

The exponential form is \( 5^{2} = 25 \).

Step 1: Identify Components of the Logarithm

The given logarithmic equation is \( \log_{5} 25 = 2 \). Here, the base of the logarithm is 5, the result of the logarithm is 2, and the number inside the log is 25.

Step 2: Understand the Exponential Form

In the context of logarithms, if \( \log_{b} a = c \), it means \( b^{c} = a \). This is the definition of a logarithm when written in its exponential form.

Step 3: Convert the Logarithmic Statement

Using the understanding from Step 2, convert \( \log_{5} 25 = 2 \) into exponential form: \( 5^{2} = 25 \). This is the equivalent exponential statement.

Key Concepts

Logarithms

Logarithms are mathematical expressions that represent the power to which a base number must be raised to obtain another number. For example, in the equation \( \log_{5} 25 = 2 \), the base is 5 and the logarithm states that 5 must be raised to the power of 2 to result in 25. This makes understanding logarithmic equations crucial in simplifying complex calculations and solving exponential equations.

• Base Number: The base in a logarithm, represented by \( b \) in \( \log_{b} a = c \), is the number that is repeatedly multiplied.
• Logarithmic Value: The result of the log equation, \( c \), signifies the exponent or the number of times the base must be multiplied by itself to reach \( a \).
• Logarithmic Argument: The \( a \) is the number generated by raising the base to the power of the exponent.

Understanding these components is key to mastering logarithmic expressions and converting them into exponential forms, thereby deepening your mathematical skills.

Exponential Form

The exponential form of a logarithmic statement offers another way of expressing the equality between numbers linked by multiplication. To translate a logarithm into its exponential form, you interpret the logarithmic equation \( \log_{b} a = c \) as \( b^{c} = a \).

• Base in Exponential Form: The base remains the same when converting to exponential form. For example, in \( 5^2 = 25 \), 5 is the base.
• Exponent: This is equivalent to the result of the logarithmic equation and represents how many times the base is multiplied by itself, such as 2 in the example \( 5^2 \).
• Resulting Number: This is the number that results from raising the base by the exponent, seen as 25 in the context \( 5^2 = 25 \).

By rewriting logarithmic equations as exponential equations, it becomes easier to handle expressions, perform calculations, and solve equations across various fields of mathematics.

Mathematics Education

In mathematics education, understanding both logarithmic and exponential forms is essential. These concepts expand a student's ability to tackle real-world applications, such as scientific calculations and complex financial models.

• Practical Understanding: Learning how to convert between logarithmic and exponential forms helps students gain a deeper comprehension of mathematical relationships and prepares them for higher-level problem-solving tasks.
• Real-world Applications: Logarithms and exponentials appear in natural phenomenon modeling, like population growth and radioactive decay, illustrating their practical importance.
• Developing Analytical Skills: By engaging with these concepts, students enhance their analytical skills, logical thinking, and mathematical intuition, all crucial for success in advanced studies and professional fields.

Incorporating these concepts effectively into education equips students with problem-solving tools that are applicable beyond the classroom, fostering a comprehensive mathematical literacy.