Problem 68 - Algebra A Combined Function

Question

Solve. $$ \log _{x} 2=-\frac{1}{3} $$

Step-by-Step Solution

Answer

The base \( x \) is \( \frac{1}{8} \).

Step 1: Understand the Logarithmic Equation

The original equation given is \( \log_{x} 2 = -\frac{1}{3} \). This means, in words, 'the logarithm of 2 in the base of \( x \) is equal to \( -\frac{1}{3} \).' Our goal is to find the base \( x \) that makes this equation true.

Step 2: Rewriting the Logarithm in Exponential Form

Recall that \( \log_{b} a = c \) can be rewritten as \( b^c = a \). Therefore, we can rewrite \( \log_{x} 2 = -\frac{1}{3} \) as \( x^{-\frac{1}{3}} = 2 \).

Step 3: Solve for \( x \)

To solve \( x^{-\frac{1}{3}} = 2 \), first rewrite it as \( \frac{1}{x^{\frac{1}{3}}} = 2 \). To isolate \( x \), take the reciprocal: \( x^{\frac{1}{3}} = \frac{1}{2} \). Next, cube both sides to solve for \( x \): \( x = \left( \frac{1}{2} \right)^{3} \).

Step 4: Calculate the Base \( x \)

Compute \( \left( \frac{1}{2} \right)^{3} \). The result is \( \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) = \frac{1}{8} \). Therefore, \( x = \frac{1}{8} \).

Key Concepts

Exponential Form

To tackle logarithmic equations, understanding how to convert between logarithmic form and exponential form is essential. This conversion is foundational because it allows us to solve equations more easily. A logarithmic equation in the form \( \log_{b} a = c \) can be re-expressed in exponential form as \( b^c = a \). This means the base \( b \) raised to the power of \( c \) will give us \( a \).

By rewriting a logarithmic equation into exponential form, we often find it much easier to solve for the unknown variable. In the initial problem \( \log_{x} 2 = -\frac{1}{3} \), converting to exponential form involves expressing it as \( x^{-\frac{1}{3}} = 2 \). This step simplifies the problem by helping us eliminate the logarithmic expression and focus on a more familiar exponential expression.

Solving Logarithmic Equations

Once you have rewritten a logarithmic equation in exponential form, solving it becomes a process of applying algebraic methods to find the unknown. For the equation \( x^{-\frac{1}{3}} = 2 \), the next logical step is to manipulate the expression to isolate \( x \).

First, by rewriting it in a more manageable form such as \( \frac{1}{x^{\frac{1}{3}}} = 2 \). Then, the goal is to isolate \( x^{\frac{1}{3}} \) by taking the reciprocal of both sides, thus obtaining \( x^{\frac{1}{3}} = \frac{1}{2} \).

Finally, you solve for \( x \) by getting rid of the fractional exponent. You can do this by cubing both sides of the equation, leading to \( x = \left( \frac{1}{2} \right)^{3} \). This approach allows us to find that \( x \) equals \( \frac{1}{8} \). The process shows how methodically applying algebraic operations unravels the solution.

Manipulating Exponents

Manipulating exponents is a critical skill when working with equations like \( x^{\frac{1}{3}} = \frac{1}{2} \). Understanding exponent rules allows for easier manipulation of the equations.

For instance, the fact that \( x^{a \cdot b} = (x^{a})^{b} \) can be used to simplify expressions. In our case, raising \( \left( \frac{1}{2} \right) \) to the third power gives us \( \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) = \frac{1}{8} \). This step is essentially computing the fraction's cube, which finishes the solution.

Additionally, knowing how to take the reciprocal and understand equivalent expressions helps when manipulating negative exponents. For example, \( x^{-\frac{1}{3}} \) being equivalent to \( \frac{1}{x^{\frac{1}{3}}} \) illustrates how altering the format can be key to solving complex equations efficiently. Mastering these exponent rules can greatly simplify the process of solving various mathematical problems.