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In mathematics, the logarithm product rule is a useful property for simplifying expressions where two logarithms with the same base are added together. Think of it as a way to "merge" or "condense" these two expressions into a single logarithm for easier handling. The rule states that if you have two logarithms with the same base, you can add them by multiplying the numbers inside the logarithms. For example, if you have \( \log_{b} a + \log_{b} c \), it is the same as \( \log_{b}(a \cdot c) \). This is particularly handy when you're trying to solve equations, like in the original exercise. In the given problem, \( \log_{3} 5 + \log_{3} x = 1 \), we apply the product rule to combine the logarithms into one, resulting in \( \log_{3}(5x) = 1 \). This step simplifies our equation, making it easier to move on to solving it.

Once we have combined the logarithms using the product rule, the next important concept is converting the logarithmic equation into exponential form. This is like translating the problem into a format that might be easier to solve. The general rule for this conversion is: if \( \log_{b} (A) = C \), then it translates to \( A = b^{C} \). This means you're expressing the logarithm as an equation involving exponents, which can often make it more straightforward to find unknown values. In our example, after applying the product rule, we have \( \log_{3}(5x) = 1 \). Changing this into exponential form gives us \( 5x = 3^{1} \), or equivalently, \( 5x = 3 \). Now it's a basic equation ready to be solved through simpler arithmetic steps.

The final step in this process is solving the equation that was translated into exponential form. Solving equations is a core mathematical skill where you manipulate the equation to find the value of an unknown variable, often denoted by \( x \). In our exercise, we have reached the equation \( 5x = 3 \). To solve for \( x \), we need to isolate it on one side. This is done by dividing both sides of the equation by 5. Doing so, we get \( x = \frac{3}{5} \). This simple arithmetic operation allows us to find the exact value of \( x \), which would be the solution to our original logarithmic equation. Understanding how to isolate variables is essential for effectively solving equations across many areas of math.