Problem 23 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{216}\).

Step 1: Understand the Logarithmic Equation

The given equation is \(\log_{6} x = -3\). We are asked to find the value of \(x\) that satisfies this equation. Recall that a logarithmic equation of the form \(log_{b} x = y\) is equivalent to the exponential form \(b^{y} = x\).

Step 2: Convert the Logarithmic Equation to Exponential Form

Using the property of logarithms, convert the logarithmic equation \(\log_{6} x = -3\) into its exponential form: \(6^{-3} = x\).

Step 3: Calculate the Exponent

Now, calculate \(6^{-3}\): \(6^{-3} = \frac{1}{6^{3}} = \frac{1}{216}\).Thus, \(x = \frac{1}{216}\).

Key Concepts

Exponential Form

A logarithmic equation such as \( \log_{b} x = y \) can be transformed into an exponential form, which offers a different perspective on the relationship between numbers. This transformation is achieved by using the basic definition of logarithms: \( b^{y} = x \). In the context of the example \( \log_{6} x = -3 \), converting it to its exponential form, we get \( x = 6^{-3} \). This tells us that \( x \) is found by raising the base \( 6 \) to the power of \( -3 \). Switching between logarithmic and exponential forms can simplify solving equations by directly showing how multiplications and divisions are related.

Properties of Logarithms

The properties of logarithms allow us to tackle various problems in algebra by simplifying complex expressions and equations. These include rules like:
- The product property, \( \log_{b}(mn) = \log_{b} m + \log_{b} n \).
- The quotient property, \( \log_{b}(\frac{m}{n}) = \log_{b} m - \log_{b} n \).
- The power property, \( \log_{b}(m^{n}) = n \cdot \log_{b} m \).
These properties are tools that help reformat complex logarithmic expressions into more manageable ones. For this simple problem, the conversion to exponential form is sufficient, but understanding these properties is crucial for solving more complicated equations that involve sums, products, or powers of logarithms.

Exponents

Exponents are a way of representing repeated multiplication of a number by itself. In the equation \( 6^{-3} \), the exponent \(-3\) signifies that we should take the reciprocal of \( 6 \) raised to the power of \( 3 \). This is calculated as:
- \( 6^{3} = 6 \times 6 \times 6 = 216 \). Therefore, - \( 6^{-3} = \frac{1}{216} \).
Exponents can be positive, which indicates multiplication, or negative, which implies division or taking the reciprocal of the positive exponent result. Understanding how to manipulate exponents correctly is vital for efficiently solving exponential equations.

Mathematical Problem Solving

Mathematical problem solving involves understanding which mathematical principles to apply and how to apply them appropriately. In solving logarithmic equations, such as \( \log_{6} x = -3 \), a key step is recognizing the need to convert from logarithmic to exponential form.
By applying knowledge of exponential forms and knowing how to evaluate exponents, we solve for \( x \) to find that \( x = \frac{1}{216} \). Each step in the process interconnects:
- Recognizing the equation form.
- Converting between equivalent forms.
- Calculating accurately using exponent rules.
These problem-solving strategies don't just apply to this specific example but can be used in a wide array of mathematical contexts where similar transformations and calculations are necessary.