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Logarithms have many useful properties that allow us to simplify and manipulate logarithmic expressions. These properties stem from the relationship between logarithms and exponents. Here are some important properties to know:

• Product Property: \(\log(ab) = \log(a) + \log(b)\)
• Quotient Property: \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\)
• Power Property: \(\log(a^b) = b \cdot \log(a)\)
• Inverse Property: \(\log\left(\frac{1}{a}\right) = -\log(a)\)

In the exercise, we used the inverse property, which helps in recognizing how logarithms can turn division into subtraction by introducing a negative sign. This specific property tells us that logarithms can express the reciprocal of a number as a negative logarithm of the number itself, making simplifications straightforward and efficient.

Exponential functions are essential in understanding logarithms, as they are two sides of the same coin. An exponential function is expressed as \(y = a^x\), with "a" being the base and "x" the exponent.
Exponential functions grow rapidly, producing values that increase or decrease rapidly depending on the sign and magnitude of the exponent.Logarithms are essentially the inverse of exponential functions. This means that if \(y = a^x\), then \(x = \log_a(y)\). Getting familiar with this inverse relationship helps a lot in simplifying and solving logarithmic expressions.
It is crucial to understand that logarithms answer the question, "To what power must the base be raised, to produce a certain number?"
In base 10, which is common for real-world applications, this relationship is direct since we're only seeking the power needed to transform 10 into any given number.

Base 10 logarithms, or "common logarithms," are frequently used in mathematics and science. They are denoted simply as "\(\log\)", without specifying the base, because the base is implicitly 10. For base 10, \(y = 10^x\) translates to \(x = \log(y)\).
A few properties make base 10 logarithms simple to use:

• The \(\log(10) = 1\), since 10 raised to the power of 1 is 10 itself.
• The \(\log(1) = 0\), because 10 raised to the power of 0 is 1.
• Values between 1 and 10 will have decimals, like \(\log(2) ≈ 0.301\).

In the exercise, we simplified \(\log\left(\frac{1}{10}\right)\) by applying these concepts, recognizing that \(\log(10)\) equals 1, and, therefore, \(-\log(10)\) resulted in -1. Base 10 logarithms are favored for calculations because they align with the decimal system we use every day.