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The change of base formula is essential when working with logarithms. It allows you to convert a logarithm of any base into a logarithm of another base. This can make solving logarithmic equations much simpler. The formula is:
\[ \text{log}_a b = \frac{\text{log}_c b}{\text{log}_c a} \]
Here, 'a' is the original base, 'b' is the number, and 'c' is the new base you are changing to. In our exercise, we used it to simplify the expression from \[ \text{log}_{4^{-1}} 16 \] to \[ \frac{1}{-1} \text{log}_4 16. \]Remember, you can choose any convenient base 'c', often base 10 or the natural logarithm base 'e' is used. This formula is particularly useful when the original base is not simple to work with.

The logarithm power rule simplifies logarithms involving exponents. According to this rule: \[ \text{log}_a (b^c) = c \text{log}_a b. \]In simple terms, you can move the exponent in front of the logarithm. In our step-by-step solution, we applied this rule to \[ \text{log}_4 (4^2). \]We simplified it to \[ 2 \text{log}_4 4. \]Because \[ \text{log}_4 4 = 1, \] the expression simplified to 2.
This rule is powerful when dealing with complex logarithmic expressions involving exponents. It helps break down the expression into simpler components, making it easier to solve.

Properties of logarithms are a set of rules that help simplify logarithmic expressions. Some of the most important properties include:

• Product Rule: \[ \text{log}_a (bc) = \text{log}_a b + \text{log}_a c \]
• Quotient Rule: \[ \text{log}_a \frac{b}{c} = \text{log}_a b - \text{log}_a c \]
• Power Rule: \[ \text{log}_a (b^c) = c \text{log}_a b \]
• Change of Base Formula: \[ \text{log}_a b = \frac{\text{log}_c b}{\text{log}_c a} \]

We used some of these properties in the exercise solution. For instance, the power rule and change of base formula made it easier to simplify the given logarithmic expression. Understanding these properties helps you quickly rewrite and solve various logarithmic expressions.

Exponent rules play a key role when working with logarithms since logarithms are inherently related to exponents. Some essential exponent rules are:

• Product of Powers: \[ a^m \times a^n = a^{m+n} \]
• Quotient of Powers: \[ \frac{a^m}{a^n} = a^{m-n} \]
• Power of a Power: \[ (a^m)^n = a^{mn} \]
• Negative Exponent: \[ a^{-n} = \frac{1}{a^n} \]

In our exercise, rules like converting \[ \frac{1}{4} \] to \[ 4^{-1} \] and recognizing that 16 can be written as \[ 4^2 \] are key exponential concepts. Mastering these rules helps in converting between different forms and simplifying logarithmic expressions effectively.