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Logarithmic equations are mathematical expressions that involve logarithms, with a common goal of solving for the variable within the equation. In the exercise provided, we are working with a logarithmic function, specifically a logarithm base of 4. The task involves finding the values of this logarithmic function at different given points: \( f(4) \), \( f(1/4) \), and \( f(8) \). A logarithmic equation generally takes the form \( \log_b x = y \), where \( b \) is the base of the logarithm, \( x \) is the argument, and \( y \) is the result or the exponent to which the base must be raised to yield \( x \). This means, if we solve for \( y \), we find out what power \( b \) must be raised to result in \( x \).
In simpler terms, solving a logarithmic equation means determining the exponent that satisfies the equation for a given base and argument.

• For \( f(4) = \log_4 4 \), since 4 raised to the power of 1 equals 4, our solution is 1.
• For \( f(1/4) = \log_4 \frac{1}{4} \), since 4 raised to the power of -1 equals \( \frac{1}{4} \), our solution is -1.
• For \( f(8) = \log_4 8 \), since 4 raised to the power of \( \frac{3}{2} \) equals 8, our solution is \( \frac{3}{2} \).

Understanding exponent rules is crucial for solving logarithmic equations. Exponents give us a way to express repeated multiplication in a shorthand form. For instance, \( b^n \) represents the base \( b \) multiplied by itself \( n \) times.
Here are a few fundamental exponent rules that are essential:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents. \( b^m \times b^n = b^{m+n} \).
• Power of a Power Rule: To raise a power to another power, multiply the exponents. \( (b^m)^n = b^{m \cdot n} \).
• Power of a Product Rule: To find a power of a product, distribute the exponent to each factor in the product. \( (xy)^n = x^n \cdot y^n \).
• Negative Exponent Rule: A negative exponent means to take the reciprocal of the number and then apply the positive exponent. \( b^{-n} = \frac{1}{b^n} \).
  These rules are used to simplify expressions and solve for unknown variables. For example, converting \( 4^{-1} \) to \( \frac{1}{4} \) demonstrates the negative exponent rule in action.

The base of a logarithm is a crucial part of understanding how logarithms work. Logarithms essentially answer the question: "To what power must the base be raised, to produce a given number?" This base is the same for an entire logarithmic function.
When we say \( \log_b x \), \( b \) is the base. It’s always a positive number, and cannot be 1. Choosing a base of 4, for instance, helps in the specific problem where all computations revolve around powers of 4.
- When we wrote \( \log_4 4 = 1 \), it meant "What power do we raise 4 to get 4?". The answer is clearly 1.- Similarly, \( \log_4 \frac{1}{4} = -1 \) means "4 raised to what power gives \( \frac{1}{4} \)?".- Lastly, in \( \log_4 8 \), we asked "4 raised to what power equals 8?" which results in \( \frac{3}{2} \).
Selecting the right base is not arbitrary. Different contexts and problems can use different bases, such as 10 in common logarithms or \( e \) for natural logarithms. Understanding the base helps in deconstructing and solving complicated mathematical problems.