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A logarithm answers the question: 'To what power must we raise a specific base to obtain a given number?' For example, \(\text{log}_2 8 \) asks, 'To what power must 2 be raised to get 8?' The answer is 3, because \(2^3 = 8\). This concept is essential in various fields such as science, engineering, and finance. Common logarithms use base 10 and are denoted as \(\text{log}\), while natural logarithms use base \(e\) and are denoted as \(\text{ln}\).Logarithms have several properties that make them useful for simplifying complex calculations:

• Product Rule: \(\text{log}_b (XY) = \text{log}_b X + \text{log}_b Y\).
• Quotient Rule: \(\text{log}_b (X/Y) = \text{log}_b X - \text{log}_b Y\).
• Power Rule: \(\text{log}_b (X^Y) = Y \text{log}_b X\).

Understanding these properties can help simplify expressions and solve logarithmic equations more easily.

Natural logarithms are a specific type of logarithm that uses the mathematical constant \(e\) (approximately 2.718) as the base. They are often written as \( \text{ln} \) instead of \( \text{log}_e \). Natural logarithms are particularly useful in calculus and mathematical modeling because the derivative and integral of \( \text{ln} x \) have simple and elegant forms.Here's why they are important:

• **Growth and Decay**: Many natural processes, like population growth and radioactive decay, are modeled using the natural logarithm because they involve continuous growth or decay.
• **Compound Interest**: Financial calculations involving continuously compounded interest also utilize natural logarithms.

Using a calculator, you can quickly find the natural logarithms of numbers. For instance, given the exercise problem, \( \text{ln} 12 \) and \( \text{ln} 3 \) were calculated as approximately 2.4849 and 1.0986, respectively.

Base conversion in logarithms allows us to change the base of a logarithmic expression to another base that is more convenient for computation. The change-of-base formula states:\[ \text{log}_a b = \frac{\text{log}_c b}{\text{log}_c a} \]Where \ a \ and \ b \ are positive numbers and \ c \ is the new base. This method is particularly helpful when calculators are involved, as they typically only provide logarithms to base 10 (common logarithm) and base \( e \) (natural logarithm).To apply this in the given exercise, we converted \( \text{log}_3 12 \) to a fraction of natural logarithms:\[ \text{log}_3 12 = \frac{\text{ln} 12}{\text{ln} 3} \]This converts our original base 3 logarithm into base \( e \), which can be easily computed using a calculator.With the values \( \text{ln} 12 \) approximately 2.4849 and \( \text{ln} 3 \) approximately 1.0986, the division yields approximately 2.262. This method is efficient and reliable for solving logarithms with unconventional bases.