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When dealing with logarithmic expressions, understanding how to simplify and manipulate them is crucial. A logarithm, written as \(\log_b(x)\), represents the power to which the base \(b\) must be raised to produce the number \(x\). For the common logarithm, the base \(b\) is usually 10, and is often written simply as \(\log(x)\) without the base specified.

In the equation \(2 \log x - \log x = 3\), the expression \(2 \log x\) can be understood as adding two logarithms: \(\log x + \log x\). To simplify this expression, notice each part has the log term of the same variable. Simplifying gives us a single logarithm: \(\log x\). Thus, the change from multiple terms to a single logarithmic term is a simplification process that utilizes the properties of logarithms.

The exponential form is incredibly useful for solving logarithmic equations. This form makes it easier to see what value a variable should take on by transitioning from a logarithmic equation to an exponentiation problem.

In this example, we start from the simplified log equation \(\log x = 3\). To convert it to exponential form, remember that \(\log x\) corresponds to \(\log_{10}x\). This means \(x\) is 10 raised to the power of 3, written as \(x = 10^3\).

This conversion highlights how logarithms and exponential functions are inverses. When you move from the logarithmic to the exponential form, you are leveraging this inverse relationship to solve for the unknown variable.

The properties of logarithms are foundational tools that make handling and solving logarithmic expressions much more straightforward. They allow you to simplify, expand, or condense logarithmic expressions to make them easier to work with.

Here are essential properties used:

• Product Property: \(\log_b(M) + \log_b(N) = \log_b(M \cdot N)\)
• Quotient Property: \(\log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right)\)
• Power Property: \(n \cdot \log_b(M) = \log_b(M^n)\)

In this particular exercise, simplifying \(2 \log x - \log x\) to \(\log x\) didn't directly apply one of these properties, but understanding the power and product properties helps see why \(2 \log x\) becomes \(\log(x^2)\) if needed. However, recognizing that like terms combined resulted back to \(\log x\) was essential to correctly simplifying before solving the equation.