Problem 8 - 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. $$ 3^{-4}=\left(\frac{1}{81}\right) $$

Step-by-Step Solution

Answer

\(\log_{3}\left(\frac{1}{81}\right) = -4\)."

Step 1: Identify the components of the exponential form

In the exponential expression \(3^{-4} = \frac{1}{81}\), identify the base, exponent, and the result. Here, the base is 3, the exponent is -4, and the result is \(\frac{1}{81}\).

Step 2: Understand logarithmic form

Logarithmic form is written as \(\log_{b}(a) = c\), where \(b\) is the base, \(a\) is the result of the exponentiation, and \(c\) is the exponent.

Step 3: Write the equation in logarithmic form

Using the components identified from the exponential form, write the expression as a logarithm: \(\log_{3}\left(\frac{1}{81}\right) = -4\).

Key Concepts

Exponential Expressions

Exponential expressions are a fundamental concept in mathematics, often seen in equations that include a base and an exponent. These expressions are written in the form of \(b^n\), where \(b\) is the base, and \(n\) is the exponent. An exponential expression represents repeated multiplication of the base. For instance, in the expression \(3^{-4}\), the base 3 is multiplied by itself, a total of four times, but in a way that produces a fractional result because the exponent is negative.
When we raise a number to a negative exponent, we are expressing its reciprocal (the one divided by the number) raised to the corresponding positive power. Thus, \(3^{-4}\) becomes \(\frac{1}{3^4}\), which simplifies to \(\frac{1}{81}\). Recognizing and manipulating exponential expressions involve understanding this powerful operation of repeated multiplication and examining the effects of different exponents, including negatives and fractions.

• Positive Exponent: A positive exponent, like \(3^4\), means 3 is multiplied by itself 4 times.
• Negative Exponent: A negative exponent, like \(3^{-4}\), indicates a division of the number. Essentially, it is the reciprocal of \(3^4\), resulting in \(\frac{1}{81}\).
• Zero Exponent: Any non-zero number raised to the power of 0 is 1, represented as \(b^0 = 1\).

Base and Exponent

The terms 'base' and 'exponent' are integral to understanding any exponential expression. The base is the number that is being multiplied by itself a specified number of times. On the other hand, the exponent indicates how many times the base is used as a factor in the multiplication.
In our example, \(3^{-4}\),

• The base is 3, which is being multiplied.
• The exponent is -4, which shows the number of times the base contributes to the multiplication in a reciprocal manner.

A clear understanding of base and exponent helps in the conversion between different mathematical forms, such as exponential and logarithmic.When converting to logarithmic form, it's crucial to accurately identify these components. In the expression, \(3^{-4} = \frac{1}{81}\), understanding the negative exponent helps convert to a form where three's role as a base remains consistent.

Logarithmic Equation

A logarithmic equation expresses the exponent that the base must be raised to produce a given number. It is the reverse operation of exponentiation. Written in the form \(\log_b(a) = c\), here, \(b\) is the base, \(a\) is the result of exponentiation, and \(c\) is the exponent.
Let's see how the conversion works using our earlier example:
- Exponential Form: \(3^{-4} = \frac{1}{81}\)
- Logarithmic Form: \(\log_3(\frac{1}{81}) = -4\)

• Base: The number we are taking the logarithm of, in this case, is 3.
• Result: What the exponentiation results in, \(\frac{1}{81}\) here.
• Exponent: The power to which the base is raised, which is -4.

Converting an exponential expression into a logarithmic equation involves clearly understanding these components and appropriately placing them in the logarithmic form. This allows us to view the relationship between numbers differently, aiding in solving many mathematical problems.