Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3$$6^{-2}=\frac{1}{36}\)

Step-by-Step Solution

Answer

The logarithmic form of the equation \(6^{-2}=\frac{1}{36}\) is \(\log_{6}\frac{1}{36} = -2\).

Step 1: Identify the Base, Exponent, and Result in the Exponential Form

In the provided exponential expression \(6^{-2}=\frac{1}{36}\), the base \(b\) is 6, the exponent \(y\) is -2, and the result \(x\) is \(\frac{1}{36}\).

Step 2: Conversion to Logarithmic Form

Translate the exponential form to logarithmic form using the definition of a logarithm. The definition of the logarithm states that if \(b^{y}=x\), then \(\log_{b}x=y\). So, replace \(b\) with 6, \(x\) with \(\frac{1}{36}\), and \(y\) with -2, to yield \(\log_{6}\frac{1}{36}=-2\).

Key Concepts

Exponential Form

In mathematics, exponential form is a way of expressing repeated multiplication of a base number. Simply put, it involves raising a number to a certain power. The formula for exponential form is represented as \( b^{y} = x \), where:

• \( b \) is the base,
• \( y \) is the exponent,
• and \( x \) is the result or value obtained from the exponentiation process.

For example, in the expression \( 6^{-2} = \frac{1}{36} \), the number 6 is raised to the power of -2, yielding a result of \( \frac{1}{36} \). The base 6 multiplied by itself -2 times signifies the multiplicative inverse, which results in a fraction because the exponent is negative. Understanding exponential form is crucial because it forms the basis for logarithmic transformations.

Base and Exponent

The terms 'base' and 'exponent' are fundamental in comprehending exponential expressions. Let's break them down:

• The base is the number that is being multiplied. It serves as the foundation for the exponential expression. In our example \( 6^{-2} \), 6 is the base.


• The exponent is the power to which the base is raised. It essentially tells us how many times the base is used in a multiplication. Exponents can be positive, negative, or zero. Here, the exponent is -2, meaning we take the reciprocal of the base raised to the positive power.

Understanding the role each component plays helps in manipulating and transforming equations effectively, such as converting them into logarithmic form.

Conversion to Logarithmic Form

Converting from exponential form to logarithmic form is a key skill in math. It allows us to express multiplication in terms of logarithms, an operation closely related to exponentiation. The fundamental rule for conversion is that if \( b^{y} = x \), then it translates to \( \log_{b}x = y \) in logarithmic form. This can be thought of as the inverse operation of exponentiation.

• Identify the base: In our example, the base \( b \) is 6.
• Pinpoint the result: \( x \) is \( \frac{1}{36} \).


• Recognize the exponent: \( y \) is \(-2\).

Putting it all together, the exponential equation \( 6^{-2} = \frac{1}{36} \) becomes \( \log_{6}\frac{1}{36} = -2 \) in logarithmic form. This conversion encapsulates how logarithms operate—relating the multiplicative power of the base to arrive at the result.