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When we encounter an exponential equation, such as \(3^x = 140\), and the other side cannot be converted into the same base (like 3 in this case), logarithms become an essential tool. A logarithm helps transform the equation from an exponential form into a more manageable algebraic form, making it easier to solve. The most common logs used are \( ext{log}_{10}\), known as the common log, and \( ext{ln}\), which is a natural logarithm with the base \(e\).
Using logarithms is beneficial because of their powerful properties, such as the ability to convert multiplicative relationships into additive ones. This is particularly helpful when dealing with exponential equations. By applying a logarithm to both sides of the equation, the equation becomes:\[ \log(3^x) = \log(140) \]
This crucial step facilitates the transition needed to use further rules of logarithms for simplification, making the equation solvable.

The power rule of logarithms is a useful property used when dealing with the exponent in a logarithmic equation. This rule states that \( \log_b(m^n) = n \cdot \log_b(m) \), where \(b\) is the base, \(m\) is the argument, and \(n\) is the exponent. This rule allows us to "bring down" the exponent as a coefficient in front of the log.
In the given example, \(\log(3^x)\) can be rewritten using the power rule, allowing us to take \(x\) out of the exponent and get:
\[ x \log(3) = \log(140) \]
This step dramatically simplifies the equation, turning an exponential equation into a linear form, which is much easier to solve. It effectively removes the difficulty of having the variable in the exponent, moving it into a coefficient that can be easily isolated through algebraic techniques.

Solving equations, specifically exponential equations using logarithms, involves a systematic approach. Once the exponent has been dealt with using the power rule, as in \(x \cdot \log(3) = \log(140)\), isolating \(x\) becomes the goal.
To solve for \(x\), divide both sides of the equation by \(\log(3)\). This step isolates \(x\), yielding:
\[ x = \frac{\log(140)}{\log(3)} \]
This equation showcases the solution of the original exponential equation. It's important to note here the role of the calculator, as computing the values of \(\log(140)\) and \(\log(3)\) can be easily done with this tool. Put these calculated values in their respective places, and you will have the numerical value of \(x\).
Understanding this method is essential since not all exponential equations can be solved by rewriting bases. Logarithms offer a versatile solution for dealing with such problems efficiently.