Problem 23 - A Graphical Approach to College Algebra

Question

Solve each equation. Give the exact answer. $$\log _{6} x=-3$$

Step-by-Step Solution

Answer

x = \( \frac{1}{216} \).

Step 1: Understand the Logarithmic Equation

We are given the equation \( \log_{6} x = -3 \). This is a logarithmic equation which asks us to find the value of \( x \) such that the base 6 raised to the power \( -3 \) equals \( x \).

Step 2: Convert Logarithmic Form to Exponential Form

To solve the equation \( \log_{6} x = -3 \), we need to convert it into exponential form. By definition, if \( \log_{b} a = c \), then \( b^{c} = a \). Therefore, \( 6^{-3} = x \).

Step 3: Calculate the Exponential Expression

Now that we have \( x = 6^{-3} \), compute the value of \( 6^{-3} \). This means we need to find the reciprocal of \( 6^{3} \). First, calculate \( 6^{3} = 6 \times 6 \times 6 = 216 \). Thus, \( 6^{-3} = \frac{1}{216} \).

Step 4: Determine the Exact Answer

We've computed \( 6^{-3} = \frac{1}{216} \), which means \( x = \frac{1}{216} \). This is the exact answer to the original equation.

Key Concepts

Exponential Form

Converting a logarithm into exponential form may sound complicated, but it's really about understanding the connection between the two forms. When you see an equation like \( \log_{6} x = -3 \), it's saying, "What power must 6 be raised to produce \( x \)?" Converting it utilizing the rule \( \log_{b} a = c \) means that \( b^{c} = a \). In our case, \( 6^{-3} = x \). The exponential form clearly tells us that whatever comes as the exponent (here, \(-3\)) is the power to which we raise the base (6) to equal the number we're looking for (\( x \)). This step is crucial because it sets the problem into a form where we can directly calculate \( x \). Remember, knowing how to switch between logarithmic and exponential forms is a powerful tool in solving logarithmic equations.

Reciprocal Calculation

Finding a negative exponent might seem tricky, but it's quite straightforward if you understand that a negative exponent suggests a reciprocal. When our exponential expression \( 6^{-3} \) appears, we interpret this as "one over \( 6^{3} \)."

• First, calculate \( 6^{3} \) by multiplying 6 three times: \( 6 \times 6 \times 6 = 216 \).
• Then, take the reciprocal of this result, which gives us \( \frac{1}{216} \).

This means \( 6^{-3} = \frac{1}{216} \). Each negative exponent changes the power into a fraction, providing a neat way to tackle equations like these. It’s all about flipping the power into a denominator, expressing it as one over the positive exponent.

Exact Answer

In many math problems, we are tasked with finding not just an answer, but an 'exact' answer. This means you must keep the answer in its most precise form. Here, converting from a logarithmic form to an exponential form led us to \( x = \frac{1}{216} \).

• This fraction is as exact as it gets since reducing fractions and simplifying numbers offers the most accurate representation.
• It's vital when working on problems requiring exact answers to avoid rounding or approximating unless explicitly told to do so.
• By using precise calculations like these, students can better understand the relationships and scaling in logarithmic equations.

Keeping the answer in the form of a fraction, especially when the calculation involves exponentiation, provides clarity and precision crucial in mathematical solutions. An exact answer maintains the essence of the problem's solution and displays your comprehension fully.