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Understanding the power rule in logarithms is crucial for simplifying expressions. The power rule states that \(\text{log}(a^b) = b \text{log}(a)\). This means that if the argument of the logarithm is raised to an exponent, we can bring the exponent in front as a coefficient.
For example, consider \(\text{log}(3^2)\). Using the power rule, this transforms into \(2 \text{log}(3)\).
By mastering this rule, you can easily handle logarithms with exponents and make complex expressions simpler.

The product rule in logarithms helps in breaking down expressions where the argument is a product of two or more factors. According to the product rule, \(\text{log}(ab) = \text{log}(a) + \text{log}(b)\).
Let's take an example: \(\text{log}(2 \times 9)\). Using the product rule, this becomes \(\text{log}(2) + \text{log}(9)\).
Notice how the product inside the logarithm splits into a sum of logarithms. This is incredibly useful when dealing with complex multiplication within logarithms.

Expressing logarithms involves using known values and rules to rewrite a logarithmic expression in its simplest form. Take the given problems \(\text{log 9}\), \(\text{log 18}\), and \(\text{log 54}\).
Using the power rule, \(\text{log 9}\) can be rewritten as \(\text{log}(3^2)\) and then \(\text{log}(3^2) = 2\text{log}(3)\).
Next, to express \(\text{log 18}\), recognize that 18 is the product of 2 and 9, so \(\text{log 18}\) becomes \(\text{log}(2 \times 9) = \text{log}(2) + \text{log}(9)\). We already know from our previous step that \(\text{log}(9) = 2\text{log}(3)\), so finally \(\text{log 18} = \text{log 2} + 2 \text{log 3}\).
Lastly, for \(\text{log 54}\), we notice 54 is the product of 2 and 27, so \(\text{log 54}\) becomes \(\text{log}(2) + \text{log}(27)\). Since 27 can be written as \(\text{log}(3^3) = 3\text{log}(3)\), then \(\text{log 54} = \text{log 2} + 3 \text{log 3}\).