- For property 8, the equality of values implies the equality of their logarithms: \(\log_a M = \log_a N\).
- For property 9, the equality of logarithms implies the equality of their arguments: \(M = N\).
- For property 10, the change of base formula is \(\log_a M = \frac{\log_b M}{\log_b a}\).
- Final answers are:
- \(\log_a M = \log_a N\)
- \(M = N\)
- \(\frac{\log_b M}{\log_b a}\)
Explanation
- Understanding the Properties of Logarithms
The problem asks us to complete three fundamental properties related to logarithmic functions. Let \(M, N,\) and \(a\) be positive real numbers with \(a
eq 1\). Logarithms are defined as the inverse of exponential functions, and they possess specific algebraic properties that allow us to manipulate expressions involving them. We will address each of these three properties systematically.
- Addressing Property 8: Equality of Logarithms
The first property, often called the equality property for logarithms, states that if two positive numbers \(M\) and \(N\) are equal, then their logarithms with respect to the same base \(a\) must also be equal. Formally, if \(M = N\), then \(\log_a M = \log_a N\). This property holds because the logarithmic function \(f(x) = \log_a x\) is a well-defined function.
- Addressing Property 9: One-to-One Nature of Logarithms
The second property addresses the injective or one-to-one nature of the logarithmic function. If \(\log_a M = \log_a N\), we can exponentiate both sides with base \(a\), leading to \(a^{\log_a M} = a^{\log_a N}\), which simplifies to \(M = N\). This ensures that distinct inputs for a logarithm must produce distinct outputs, provided the base \(a\) is the same.
- Addressing Property 10: Change of Base Formula
The final property is known as the change of base formula. This is a powerful tool that allows us to evaluate or simplify logarithms in any base by converting them into a common base \(b\). The formula is \(\log_a M = \frac{\log_b M}{\log_b a}\), where \(b\) is a new positive base not equal to 1. This formula is essential for using calculators, which typically only compute logarithms in base 10 or base \(e\).
Examples
Logarithms are used in many scientific fields to describe phenomena that occur over a wide range of scales. For instance, the Richter scale measures the intensity of earthquakes using a base-10 logarithmic scale. If you needed to convert a measurement from a natural logarithm (base \(e\)) to a common logarithm (base 10) to interpret data in terms of intensity, you would use the change of base formula: \(\ln M = \frac{\log_{10} M}{\log_{10} e}\). Understanding these properties allows you to work seamlessly between different logarithmic scales and solve exponential equations.