91Ó°ÊÓ

Logarithms follow several key properties that are essential for simplifying expressions and solving equations. One primary property is the product rule: \[\text{log}_b(mn) = \text{log}_b(m) + \text{log}_b(n)\]. This property allows you to split logarithms of products into simpler terms. The quotient rule is another vital property: \[\text{log}_b\frac{m}{n} = \text{log}_b(m) - \text{log}_b(n)\]. It helps to deal with logarithms of fractions by breaking them down into individual logs. One particularly useful property in solving equations is the power rule: \[\text{log}_b(m^n) = n \text{log}_b(m)\]. This rule enables you to move the exponent in the argument out as a multiplicative factor, making the expression easier to handle.

Simplifying logarithmic expressions often involves using the properties of logarithms to transform complex expressions into simpler forms. For instance, in the given problem, we used the power rule extensively. The innermost logarithm, \(\text{log}_2(2^{4^{\frac{4}{3}}})\), simplifies to \(4^{\frac{4}{3}}\) because the base of the logarithm and the argument are the same. Then, we apply the power rule again when simplifying \(\text{log}_2(4^{\frac{4}{3}})\). Recognizing that \(4 = 2^2\), we rewrite the expression to apply the power rule: \(4^{\frac{4}{3}} = (2^2)^{\frac{4}{3}} = 2^{\frac{8}{3}}\). Hence, \(\text{log}_2(2^{\frac{8}{3}})\) becomes \(\frac{8}{3}\).

Nested logarithms are logarithmic expressions within other logarithms, like \(\text{log}_a(\text{log}_b(x))\). Solving these requires working from the inside out. First, simplify the innermost logarithm before moving to the next. In our problem, we had multiple layers: \(\text{log}_{2}(\text{log}_{2}(\text{log}_{2}(2^{4^{\frac{4}{3}}})))\). We started by simplifying the innermost expression, \(\text{log}_{2}(2^{4^{\frac{4}{3}}})\), to get \(4^{\frac{4}{3}}\). Then we moved to the next layer: \(\text{log}_{2}(4^{\frac{4}{3}})\) and further reduced it to \(\text{log}_{2}(2^{\frac{8}{3}}) = \frac{8}{3}\). The final step involved simplifying the outermost logarithm using properties or other techniques.

The change of base formula is incredibly useful when dealing with logarithms, especially if the base is not convenient to work with. The formula is: \(\text{log}_b(a) = \frac{\text{log}_c(a)}{\text{log}_c(b)}\), where \(c\) is a new base, often chosen as 10 or \(e\). This formula helps convert logarithms from one base to another, simplifying calculations, especially on calculators that typically only allow log base 10 or natural log (base \(e\)). Although this formula wasn't directly used in our problem, it's vital to understand as it could be required in similar contexts. For our nested logarithms problem, knowing this could offer alternative ways to simplify the steps or verify results.