Problem 43 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$\log _{1 / 6} 36$$

Step-by-Step Solution

Answer

\( x = -2 \)

Step 1: Convert to Exponential Form

Remember, the logarithmic expression \( \log_{a}(b) = c \) can be converted to the exponential form \( a^{c} = b \). For our expression, we have \( \log_{1/6}(36) = x \). This means that \((1/6)^x = 36\).

Step 2: Apply the Exponent Rule

The equation \((1/6)^x = 36\) can be rewritten using the exponent rule of fractions, which says \((a/b)^n = a^n / b^n\). Here \((1/6)^x = 1^x / 6^x = 36\). Since \(1^x = 1\), we have the equation \(1 / 6^x = 36\).

Step 3: Reciprocate and Solve for Positive Exponent

Reciprocating gives \(6^x = 1/36\). To express \(1/36\) as a power of 6, note that \(36 = 6^2\), hence \(1/36 = 6^{-2}\). Now we have \(6^x = 6^{-2}\).

Step 4: Equate Exponents

Since the bases are the same, we can equate the exponents: \(x = -2\).

Step 5: Verify the Solution

Verify that substituting \(x = -2\) back into the logarithmic function makes sense. \( (1/6)^{-2} = 6^2 = 36\), which confirms our solution is correct.

Key Concepts

Exponential Form

The exponential form is a way of expressing a mathematical relationship involving growth or decay through exponents. This form is particularly useful when working with logarithms, as it translates the logarithmic expression into a format that's easier to manipulate and solve.

When you have a logarithmic equation like \( \log_{a}(b) = c \), it can be expressed in exponential form as \( a^{c} = b \). This translation helps us understand the relationship between the base \( a \), the result \( b \), and the exponent \( c \).

In our original exercise, the expression \( \log_{1/6}(36) = x \) is converted to exponential form, resulting in the equation \((1/6)^x = 36\).

This conversion is a vital step because it sets the stage for applying further rules and techniques, making solving the problem more straightforward.

Exponent Rule

Exponent rules are fundamental when dealing with powers and exponents. They help simplify expressions that involve repeated multiplication of the same base. One important rule is the power of a quotient rule:

• \( (a/b)^n = a^n / b^n \)

In our exercise, this rule plays a crucial role in simplifying the exponential expression.

For \( (1/6)^x = 36 \), applying the exponent rule gives us \( 1^x / 6^x = 36 \). Realizing that \( 1^x = 1 \) for any real number x, we simplify the equation further to \( 1 / 6^x = 36 \).

This transformation illustrates how exponent rules can break down the problem into more manageable pieces, allowing us to tackle complex expressions by simplifying them step-by-step.

Simplifying Expressions

Simplifying expressions is the process of making a mathematical expression as compact and straightforward as possible. In the context of the exponential form and exponent rules, simplification often involves reducing expressions to a single term or a more easily understandable form.

Once we have the expression \( 1/ 6^x = 36 \), reciprocating gives \( 6^x = 1/36 \). Recognizing that 36 can be expressed as \( 6^2 \), we further rewrite \( 1/36 \) as \( 6^{-2} \).

This simplification allows us to equate the exponents because the bases are now the same: \( 6^x = 6^{-2} \).

By equating the exponents, we find that \( x = -2 \), completing the simplification process. Verifying with the original expression confirms the solution, demonstrating the power of simplifying complex logarithmic expressions into something digestible and concise.