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When differentiating logarithmic functions, we often use the natural logarithm (base \(e\)), because it's simpler to work with. For any logarithmic function of the form, \(\text{log}_a x\), it can be converted to natural logarithm using the change of base formula: \(\text{log}_a x = \frac{\text{log} x}{\text{log} a}\). This makes it easier to differentiate because the derivative of the natural logarithm, \(\text{log}_e x\) or simply \(\text{ln} x\), is straightforward: \(\frac{d}{dx} (\text{ln} x) = \frac{1}{x}\). \ In the given exercise, we have \(\text{log}_8 x\). We change the base from 8 to the natural logarithm: \(\text{log}_8 x = \frac{\text{log} x}{\text{log} 8}\). Then, we can differentiate it easily: \(\frac{d}{dx} \bigg(\frac{\text{log} x}{\text{log} 8}\bigg) = \frac{1}{x \text{log} 8}\). This simplifies our work significantly.

Polynomials are perhaps the simplest functions to differentiate. For a polynomial term \(x^n\), where \(n\) is a constant, the derivative is given by: \(\frac{d}{dx} (x^n) = nx^{n-1}\). This means you bring the exponent down in front of the \(x\) and subtract one from the exponent. For example, in the exercise, we have \(u = x^3\). Applying the rule: \(\frac{d}{dx} (x^3) = 3x^2\). \ This rule works for any polynomial term, whether it's \(x^2\), \(x^5\), or any other power. It's a key building block for more complex differentiation, where polynomial terms often appear in larger expressions.

The change of base formula for logarithms is very useful for differentiation and solving logarithmic equations. It states: \(\text{log}_a b = \frac{\text{log} b}{\text{log} a}\), where the base in the denominator can be any base you prefer. \Typically, we change to base \(e\), the natural logarithm (\(\text{ln}\)), because its properties simplify differentiation and integration. For instance, in the exercise, we convert \(\text{log}_8 x\) to \(\text{ln}\) as follows: \(\text{log}_8 x = \frac{\text{log} x}{\text{log} 8}\). After this conversion, we can easily apply the derivative rule: \(\frac{d}{dx} (\text{ln} x) = \frac{1}{x}\). This results in: \(\frac{d}{dx} (\frac{\text{log} x}{\text{log} 8}) = \frac{1}{x \text{log} 8}\). \Understanding how to use and apply the change of base formula not only simplifies calculations but also gives you flexibility in dealing with different bases in logarithmic functions.