Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Give the exact answer. $$\log _{4} x=-\frac{1}{6}$$

Step-by-Step Solution

Answer

\( x = 2^{-1/3} \)

Step 1: Understand the Logarithm Equation

The given equation is \( \log_{4} x = -\frac{1}{6} \). This means that we are searching for a number \( x \) such that the base 4 logarithm of \( x \) equals \(-\frac{1}{6}\).

Step 2: Convert Logarithm to Exponential Form

To convert the logarithmic equation into an exponential equation, use the definition of logarithms: for \( \log_{b} a = c \), it is equivalent to \( b^{c} = a \). For our equation, \( 4^{-\frac{1}{6}} = x \).

Step 3: Simplify the Expression

Calculate \( 4^{-\frac{1}{6}} \). This is the same as \( \frac{1}{4^{\frac{1}{6}}} \). Since \( 4 = 2^{2} \), it further simplifies to \( 2^{-\frac{1}{3}} \).

Step 4: Express the Answer

Thus, the expression for \( x \) is \( x = 2^{-1/3} \). This is the exact answer to the equation given.

Key Concepts

Exponential Form

When we talk about exponential form, we are essentially converting an equation from one mathematical expression to another equivalent form. It's common in mathematics where we deal with powers and roots. In the context of logarithmic equations, converting to exponential form is a crucial step.
To convert a logarithmic expression into an exponential one:

• Start with the given equation \( \log_{b} a = c \).
• According to the definition of logarithms, this can be rewritten as \( b^{c} = a \).

This means if we know the base (\(b\)), exponent (\(c\)), and the result (\(a\)), we can switch between these forms.
In our example, \( \log_{4} x = -\frac{1}{6} \) becomes \( 4^{-\frac{1}{6}} = x \) once converted to exponential form. This transformation helps in simplifications and calculations.

Properties of Exponents

Exponents have some important properties that allow us to simplify expressions effectively. Understanding these can help greatly in solving equations and in manipulating algebraic expressions.
Some of the key properties of exponents include:

• \( a^{m} \times a^{n} = a^{m+n} \) - When multiplying like bases, add the exponents.
• \( \frac{a^{m}}{a^{n}} = a^{m-n} \) - When dividing like bases, subtract the exponents.
• \( (a^{m})^{n} = a^{m\cdot n} \) - When raising an exponent to another power, multiply the exponents.
• \( a^{-m} = \frac{1}{a^{m}} \) - A negative exponent means reciprocal.
• \( a^{0} = 1 \) - Any non-zero base raised to the zero power is 1.

In the solution provided, converting \( 4^{-\frac{1}{6}} \) involves using the negative exponent property: \( 4^{-\frac{1}{6}} = \frac{1}{4^{\frac{1}{6}}} \). Then, recognizing that \( 4 = 2^{2} \), we can apply further exponent rules to simplify.

Definition of Logarithms

Logarithms are the inverse operation of exponentiation. This means they help us find which power a number (called the base) must be raised to, to arrive at another number.
For example, if \( a^{b} = c \), then \( \log_{a} c = b \). This definition is pivotal for rewriting logarithmic equations into exponential form, aiding in problem-solving.
Logarithms have some unique properties:

• \( \log_{a}(xy) = \log_{a}(x) + \log_{a}(y) \) - The logarithm of a product is the sum of the logarithms.
• \( \log_{a}\left(\frac{x}{y}\right) = \log_{a}(x) - \log_{a}(y) \) - The logarithm of a quotient is the difference.
• \( \log_{a}(x^{n}) = n\log_{a}(x) \) - The logarithm of a power is the exponent times the logarithm.
• \( \log_{a}(1) = 0 \) - The logarithm of 1 is always 0.
• \( \log_{a}(a) = 1 \) - The logarithm of the base is always 1.

In the original problem, \( \log_{4} x = -\frac{1}{6} \), using the definition of a logarithm allows us to express it as \( 4^{-\frac{1}{6}} = x \), simplifying our path to the solution.