Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Express each equation in logarithmic form. $$32^{3 / 5}=8$$

Step-by-Step Solution

Answer

The logarithmic form of the equation \(32^{\frac{3}{5}} = 8\) is: \( \log_{32}(8) = \frac{3}{5} \).

Step 1: Identify the base, exponent, and result

In the given equation, we have: _base_ (b): 32 _exponent_ (x): \(\frac{3}{5}\) _result_ (y): 8 Now that we have identified these values, we can rewrite the equation in logarithmic form.

Step 2: Rewrite the equation in logarithmic form

Utilizing the formula we mentioned earlier, the logarithmic form of the given equation can be expressed as: \(\log_b(y) = x\) In this specific case, we have: \(\log_{32}(8) = \frac{3}{5}\) So the logarithmic form of \(32^{3 / 5} = 8\) is: $$\log_{32}(8) = \frac{3}{5}$$

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number. They consist of a base and an exponent. The base is the number that is being multiplied, while the exponent indicates how many times the base is multiplied by itself. For example, in the expression \(32^{3/5}\), the base is 32, and the exponent is \(\frac{3}{5}\). This fractional exponent means we are dealing with both root and power operations.

• The numerator of the exponent (3) indicates the power, meaning we take the cube of the root operation.


• The denominator (5) suggests taking the fifth root of the base.

Both operations combine to transform 32 in such a way that results in 8. Understanding how exponents work with fractional values allows us to explore the flexibility and depth of powers in mathematics.

Deciphering Logarithms

Logarithms are the inverse operations of exponentiation. While exponents deal with multiplying a number by itself, logarithms answer the question: "To what power should we raise a certain base to obtain a specific number?" In the context of the provided equation \(32^{3/5} = 8\), we express the same mathematical relationship using logarithms.

• The base of the exponent becomes the base of the logarithm.


• The result of the exponentiation becomes the number inside the logarithm.


• The exponent itself becomes the result of the logarithmic expression.

Hence, converting \(32^{3/5} = 8\) into logarithmic form, we get \(\log_{32}(8) = \frac{3}{5}\). This form allows us to interpret exponential relationships in a different and often more insightful way.

Equation Transformation: From Exponential to Logarithmic Form

Equation transformation involves rewriting a mathematical expression into a different form that may be more useful for solving a problem. In the case of transforming an exponential equation like \(32^{3/5} = 8\) to a logarithmic one, the process involves switching the components as described in the logarithmic concept.
Breaking it down:

• Identify the base (b), which, in this case, is 32.


• Determine the exponent (x), as \(\frac{3}{5}\).


• Establish the result (y), which is 8.

By recognizing these parts, we can write the equation in logarithmic terms. This highlights the beauty of equation transformation, as it allows for calculations that might otherwise be hidden or complex, making them more straightforward and accessible.