Problem 8 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Express each equation in logarithmic form. $$81^{3 / 4}=27$$

Step-by-Step Solution

Answer

The logarithmic form of the given equation \(81^{3/4} = 27\) is \(\log_{81} 27 = \frac{3}{4}\).

Step 1: Identify the base, exponent, and result

In the given equation, \(81^{3/4} = 27\), the base is 81, the exponent is 3/4, and the result is 27.

Step 2: Convert the equation to logarithmic form

To convert the exponential equation to a logarithmic equation, we'll use the following rule: If \(b^x = r\), then \(\log_b r = x\), where b is the base, x is the exponent, and r is the result. Applying this rule to our equation, we have: \(\log_{81} 27 = \frac{3}{4}\). So, the logarithmic form of the given equation is: \[\log_{81} 27 = \frac{3}{4}\]

Key Concepts

Base of Logarithm

The base of a logarithm is fundamental to understanding logarithms. It's the number that is raised to a given power to obtain another number. In the problem we are solving, the base is 81 in the equation \(81^{3/4} = 27\).
Talking more broadly:

• The base in such mathematical expressions is a crucial component that defines how numbers are transformed or expressed.
• A logarithm answers the question: "To what exponent must the base be raised, to produce a particular number?" In our example, it helps us relate 81 to 27 through the exponent \(\frac{3}{4}\).

Understanding the base makes converting between different forms of equations, such as exponential and logarithmic forms, much clearer and less daunting.

Exponentiation

Exponentiation is the mathematical operation that involves raising a base number to a power or exponent. This concept is central to many areas of mathematics, including logarithms. In our equation \(81^{3/4} = 27\), 81 is the base and 3/4 is the exponent.

• Exponentiation allows us to express multiplication of a number by itself multiple times succinctly. For example, \(2^3\) signifies \(2 \times 2 \times 2\).
• The exponent in the expression indicates how many times the base is used in the multiplication, or, as in our problem, a fractional exponent like \(\frac{3}{4}\) involves taking roots as well as powers.

Grasping exponentiation makes solving logarithmic equations smoother, as it directly relates to how you express numbers in different forms.

Logarithm Conversion

Changing an exponential equation to a logarithmic equation involves applying a simple but essential transformation. This is known as logarithm conversion.
In the equation \(81^{3/4} = 27\), we convert it to the logarithmic form to make it easier to understand the relationships between the numbers involved.

• The rule is: if you have \(b^x = r\), then in logarithmic form, it is \(\log_b r = x\).
• This means for our problem, \(\log_{81} 27 = \frac{3}{4}\).

Logarithm conversion is a handy tool in many mathematical fields. It helps simplify calculations, especially where exponents are complex, offering a clearer perspective by hiding messy multiplications within the concise form of a logarithm.

Mathematical Notation

Mathematical notation is a powerful language that conveys complex information succinctly and precisely. In the case of logarithmic equations, notation is especially crucial. It enables better understanding and communication of ideas like those in our example, \(\log_{81} 27 = \frac{3}{4}\).

• Notation lets mathematicians create universally recognized expressions, reducing misunderstandings.
• In logarithms, this involves indicating the base and the result clearly with subscripts and placements, which ensures consistent interpretation of the equation.

Understanding mathematical notation means you can read, write, and interpret mathematical expressions with confidence, ultimately aiding in learning more advanced concepts by building a strong foundational understanding.