Problem 8 - Algebra for College Students

Question

Write each exponential statement in logarithmic form. For example, \(2^{5}=32\) becomes \(\log _{2} 32=\) 5 in logarithmic form. (c) \(\left(g^{-1} \circ f^{-1}\right)(x)\) \(\frac{x+7}{6}\) $$ 3^{-4}=\frac{1}{81} $$

Step-by-Step Solution

Answer

\( \log_3{\left(\frac{1}{81}\right)} = -4 \)

Step 1: Identify the Parts of the Exponential Form

In the exponential statement, identify the base, exponent, and the result. For the given problem, the base is 3, the exponent is -4, and the result is \( \frac{1}{81} \).

Step 2: Write the Logarithmic Form

In logarithmic form, the equation \( b^y = x \) becomes \( \log_b{x} = y \). For our statement \( 3^{-4} = \frac{1}{81} \), this translates to \( \log_3{\left(\frac{1}{81}\right)} = -4 \).

Key Concepts

Exponential Form

Exponential form is a way of representing repeated multiplication of a number by itself. It consists of a base and an exponent. The base is the number you are multiplying, while the exponent is the number of times you multiply the base by itself. For example, in the exponential expression \( 3^{-4} \), the base is 3 and the exponent is -4. This expression can be expanded to mean you divide 1 by the base 4 times, which results in \( \frac{1}{81} \). This is because \( 3^{-4} = 1 / (3 \times 3 \times 3 \times 3) \). Understanding exponential form is crucial as it is the starting point for identifying components needed for transforming into logarithmic form.

• Base: The number that is multiplied by itself.
• Exponent: The power to which the base is raised, indicating repeated multiplication or division.
• Result: The final value after applying the exponential operation.

Logarithm Properties

Logarithms are the inverse operations of exponentials. They help us solve equations where the unknown is an exponent. The essential properties of logarithms include the product, quotient, and power rules, which are derived from the properties of exponents.

• Product Rule: \( \log_b{(xy)} = \log_b{x} + \log_b{y} \)
• Quotient Rule: \( \log_b{\left(\frac{x}{y}\right)} = \log_b{x} - \log_b{y} \)
• Power Rule: \( \log_b{(x^y)} = y \cdot \log_b{x} \)

These rules allow us to manipulate logarithmic equations and expressions efficiently. For example, when converting an exponential form like \( 3^{-4} = \frac{1}{81} \) into logarithmic form, we apply the understanding that the log base 3 of 1/81 equals -4, illustrating the inverse relationship of exponentiation and logarithms.

Base of a Logarithm

In a logarithmic expression, the base is the same as in its corresponding exponential form. The base tells us the number that we repeatedly multiply or divide by itself. For instance, with the equation \( \log_3 \left( \frac{1}{81} \right) = -4 \), the base is 3. This signifies that 3 raised to the power of -4 gives us \( \frac{1}{81} \).Choosing the correct base is vital because it defines the relationship between the numbers in an equation. Changing the base changes the entire problem, much like using a different unit of measure would. Thus, in transforming the exponential form \( 3^{-4} = \frac{1}{81} \) into its logarithmic equivalent, \( \log_3 \), we maintain consistency by utilizing the same base. This consistency is key in ensuring calculations remain accurate and solves the problem effectively. Always remember that logarithms can simplify complex multi-step problems, where the base connects all pieces like a common thread.