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Understanding the basics of logarithms is crucial when studying algebra and solving equations that involve exponential relationships. A logarithm is simply another way to express an exponential equation. Consider it as asking the question, 'To what power do we need to raise the base to obtain a certain number?'

In any logarithmic expression, such as \( \log_b(y) = x \), 'b' is referred to as the base, 'y' is the number we are taking the logarithm of, and 'x' is the logarithm itself, which tells us the power that 'b' must be raised to, to get 'y'. For our exercise, the logarithmic form of \(100 = 10^2\) translates to \(\log_{10}(100) = 2\), which simplifies to asking the question, 'To what exponent must 10 be raised, to result in 100?' And the answer is 2.

Furthermore, the concept of 'logs' comes with certain properties that can simplify complex expressions, including the product, quotient, and power rules which can make computations easier. Logs are the inverse functions of exponentials, which is why converting between the two forms is an essential skill in mathematics.

Exponential equations feature variables in the exponent, such as \(b^x = y\), where 'b' is the base and 'y' is the result when 'b' is raised to the power of 'x'. They describe growth or decay processes, like population growth, radioactive decay, and interest in finance. Fundamentally, they tell us how a quantity multiplies over time.

Unlike linear equations, where the variable is raised to the power of 1, exponential equations can have much more dramatic effects due to their increasing (or decreasing) rates. Solving exponential equations typically involves isolating the variable in the exponent, and this is where our knowledge of logarithms becomes incredibly handy, as logarithmic functions are used to undo exponential functions. In the given exercise, \( 10^2 = 100 \) is an exponential equation since the variable 2 is the exponent of the base 10.

Logarithmic equations, such as \(\log_b(y) = x\), contain logarithms with variables either in the base, the argument ('y' in this case), or the value of the logarithm ('x'). They are often used to solve exponential equations since taking the log of both sides of an exponential equation typically allows us to solve for the variable in the exponent.

When working with logarithmic equations, one of our goals is to rewrite the equation in a form that makes it easier to solve. This often entails using properties of logarithms, such as those mentioned earlier, or converting the logarithmic equation to its exponential form, which can be more straightforward to solve. For the exercise at hand, writing the exponential equation \(100 = 10^2\) as a logarithmic equation \(\log_{10}(100) = 2\) gives a clear indication of the power relationship between 10 and 100, and is a necessary step for solving more complex problems involving logarithms.