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Logarithms are a powerful tool in mathematics, mostly used to solve exponential equations. A logarithm answers the question: 'To what exponent must a base be raised to produce a given number?' For example, the logarithm base 10 of 100 is 2, written as \(\text{log}_{10}(100) = 2\), because 10 raised to the power of 2 is 100. In natural logarithms, we use base e (approximately 2.718). The natural logarithm is denoted as \( \text{ln} \).
When dealing with exponential decay equations, logarithms help us isolate the variable in the exponent. For instance, to solve \((0.914)^{t}=0.7019\), we take the natural logarithm of both sides to obtain the equation \(\text{ln}(0.7019) = t \text{ln}(0.914)\). This simplification follows from the power rule, which states that \(\text{ln}(a^b) = b \text{ln}(a)\). By applying this rule, we can solve for the time variable \( t \). Understanding these properties of logarithms is key for solving exponential decay problems effectively.

Exponential functions are mathematical functions that grow or decay at a constant rate per unit interval. They can be expressed in the form \ (P(t) = P_0 a^t) \ where \( P_0 \) is the initial amount, \ (a) \ is the base representing the growth or decay factor, and \( t \) is the variable for time.
In the given exercise, the function \( P(t)=21.4(0.914)^t \) is used to model the exponential decay of smokers who remain smoke-free with telephone counseling. Here, \ (21.4) \ represents the initial percentage of smokers who quit, and \ (0.914) \ is the decay factor, indicating a decrease in the percentage of non-smokers over time.
Exponentials play a crucial role in various fields like biology, finance, and physics as they describe processes that evolve over time, either growing or decaying exponentially. The key property is that the rate of change is proportional to the current value, rendering the function consistent and predictable in behaviors modeled by it.

Solving equations, particularly exponential equations, often involves several steps including manipulating the equation and using logarithms to isolate the variable. Here is some detailed advice on how to approach such problems:

• Identify the structure of the equation. For exponential equations, look for forms like \ P(t) = P_0 a^t\.
• Express the exponential part in a manner suitable for applying logarithms. For instance, converting \( a^t \) to \ \text{ln}(a^t) \ by taking the natural logarithm of both sides.
• Use logarithmic properties to simplify the equation. For example, apply \ \text{ln}(a^b) = b \text{ln}(a) \ to bring the exponent down.
• Isolate the variable, making sure to carefully perform algebraic operations on both sides.

In the provided exercise, this process was carried out efficiently. We took natural logarithms to solve for \( t \), transforming the exponential form into a linear equation for easier manipulation. Always verify your results by plugging back into the original equation to ensure consistency.