Problem 1 - Intermediate Algebra

Question

For Problems \(1-10\), write each of the following in logarithmic form. For example, \(2^{3}=8\) becomes \(\log _{2} 8=3\) in logarithmic form. $$ 2^{7}=128 $$

Step-by-Step Solution

Answer

The logarithmic form is \(\log_{2} 128 = 7\).

Step 1: Identify the given equation in exponential form

We have the equation given as:\[2^7 = 128\] Here, the base of the exponential equation is 2, the exponent is 7, and the result is 128.

Step 2: Understand the exponential to logarithmic form conversion

In general, if we have an equation in the form of \(b^c = a\), its equivalent logarithmic form would be \(\log_{b} a = c\).

Step 3: Apply the conversion to the given problem

Following the concept of converting an exponential equation to logarithmic form, the equation \(2^7 = 128\) changes as follows:The base \(b\) is 2, the number \(a\) is 128, and the exponent \(c\) is 7. So, we write it in logarithmic form as:\[\log_{2} 128 = 7\]

Step 4: Verify the conversion

To ensure that the conversion is correct, check if raising 2 (the base) to the power of 7 results in 128:\[2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128\]The conversion is verified.

Key Concepts

Exponential Functions

Exponential functions are an essential type of mathematical function, typically expressed in the form \(b^c = a\), where \(b\) is the base, \(c\) is the exponent, and \(a\) is the result. These functions model situations where a quantity grows or decays at a constant rate per unit interval. It is critical to understand how they function since exponential growth and decay appear in various fields like population studies, finance, and physics.

In our specific exercise, we explored \(2^7 = 128\). Here, the base \(b\) is 2, indicating the factor by which the quantity multiplies in each step or unit interval. The exponent \(c\) is 7, demonstrating how many times we multiply the base by itself to get the result. This kind of growth is very rapid, even starting from modest bases like 2.

• Exponential functions have a constant ratio of change, unlike linear functions.
• They can have any positive base, but a base greater than 1 is common for growth scenarios.

Understanding exponential functions allows one to easily transition into their inverse, logarithmic functions, which are integral in numerous mathematical computations.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. Essentially, if an exponential equation is given as \(b^c = a\), converting this into a logarithmic form would appear as \(\log_b a = c\). The log function helps in determining the exponent needed to achieve a particular result with a given base.

To convert from an exponential to a logarithmic function, identify the base, the result, and the exponent from the equation. For \(2^7 = 128\), the logarithmic form is \(\log_2 128 = 7\). Here, the base 2 becomes the base in the logarithm, \(128\) is the result we are taking the logarithm of, and \(7\) is the exponent that becomes the result of the logarithmic function.

• Logarithms are useful for solving equations involving exponential functions.
• They simplify multiplication and division into addition and subtraction, respectively, making complex arithmetic more convenient.
• Common logarithms (base 10) and natural logarithms (base \(e\)) are standard in scientific and engineering calculations.

By mastering logarithmic functions, students can effectively analyze exponential relationships across multiple disciplines.

Mathematical Notation

Mathematical notation serves as a universal language for expressing mathematical concepts and relationships clearly and concisely. In dealing with exponential and logarithmic functions, specific notation is imperative to avoid ambiguity.

For example, in exponential form \(b^c = a\), the notation dictates keeping the base and the exponent clearly separated, with the exponent raised and to the right of the base. Similarly, in logarithmic form \(\log_b a = c\), the base \(b\) is written as a subscript. It is important in mathematical communication to adhere to these standards to ensure precision and clarity.

• Proper notation prevents misunderstandings in complex calculations and representations.
• It facilitates the sharing and interpretation of mathematical results globally.
• Learning the correct usage of notation is essential in ensuring effective problem-solving and analysis.

Understanding and consistently using proper mathematical notation augments a student's ability to engage with and solve mathematical problems efficiently, including those involving transformations between exponential and logarithmic forms.