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A logarithm is a mathematical function that helps us understand exponents. It tells us what power we need to raise a given number (called the base) to obtain another number. The most common logarithm used in problems is the base 10 logarithm, also known as the common logarithm. We often write this as \( \log x \), which refers to \( \log_{10} x \). For example, if \( \log_{10} 100 = 2 \), that means \( 10^2 = 100 \).
Logarithms have special properties that make manipulating equations involving them easier. These include:

• The product rule: \( \log(ab) = \log a + \log b \).
• The quotient rule: \( \log(a/b) = \log a - \log b \).
• The power rule: \( \log(a^b) = b \cdot \log a \).

This makes logarithms particularly useful in solving exponential equations, as they help isolate the variable from the exponent.

Solving equations is about finding values of the variables that make the equation true. When dealing with logarithmic equations, we often use properties of logarithms to resolve such equations. In our problem \( \log (\log x) = 2 \), using the power of 10 helps simplify the equation to a familiar form.
We know \( \log x = 100 \) because \( 10^2 = 100 \). From here, the equation is resolved by understanding that the logarithm of \( x \) is now a straightforward addition problem. Furthermore, to solve for \( x \), we expel the logarithm by exponentiating both sides. This means we raise 10 to the power of \( \log x = 100 \), resulting in \( x = 10^{100} \).

This process highlights how converting logarithmic statements into exponential ones can make solving such problems more intuitive and manageable.

Mathematical reasoning involves using logic to solve problems systematically. In working with equations, it requires an understanding of how various mathematical operations interact.
For logarithmic equations, this means understanding the basic properties of logarithms and how they can simplify complex expressions. When addressing the problem \( \log (\log x) = 2 \), mathematical reasoning allows us to deduce the steps necessary for simplification.
Recognizing that \( \log(\log x) = 2 \) can transform into \( \log x = 100 \) using powers of 10 showcases a profound insight gained from logically manipulating the initial equation.

• First, we grasp the implication of the logarithmic equality, understanding the need to ‘undo’ the outer logarithm.
• Second, interpreting the result to finally solve for \( x \) emphasizes the interplay of different mathematical concepts such as exponentiation and simplification.

Effective reasoning consolidates foundational knowledge of algebraic operations, enabling the solution of more complex problems with confidence and clarity.