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A logarithm is a mathematical operation that helps us find the power to which a number, called the base, must be raised to produce another number. When the base is not explicitly mentioned, it is generally assumed to be 10, which is known as the common logarithm. This is particularly common in scientific calculations.

This means when you see \( \log x \), it is equivalent to \( \log_{10} x \). For example, \( \log 10 \) means what power do we raise 10 to get 10? The answer is 1, because 10 to the power of 1 is 10. This forms the identity \( \log_{10} 10 = 1 \).

Using base 10 makes calculations easier because it connects directly to decimal system, which is our most frequently used numerical system.

Logarithms have several useful properties that simplify mathematical expressions and calculations. Learning these properties makes it easier to manipulate and solve equations involving logarithms.

• Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• Quotient Rule: \( \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) \)
• Power Rule: \( \log_b(x^n) = n \cdot \log_b(x) \)

The quotient rule is particularly important for our example, \( \log \frac{1}{10} \). We can express this as \( \log(1) - \log(10) \). The product, quotient, and power rules are inherited from the properties of exponents, showing a deep connection between logarithms and exponential functions.

Evaluating logarithms often involves applying the properties of logarithms and understanding the base. In our problem \( \log \frac{1}{10} \), we start by using the quotient rule of logarithms.

Through the expression \( \log(1) - \log(10) \), we aim to compute each log separately:

• The logarithm of 1 is always 0, irrespective of the base. This is because any base raised to the 0 power equals 1.
• For the log base 10 of 10, the answer is 1 because 10 to the power of 1 is 10, as previously explained.

Once the values are computed, we subtract them: \( 0 - 1 \). Therefore, the evaluated logarithm for \( \log \frac{1}{10} \) is \(-1\). Understanding this process helps improve skills in solving logarithmic equations manually without needing a calculator.