91Ó°ÊÓ

Logarithms are the inverse operations of exponents, much like subtraction is to addition. In simple terms, if you know that an exponent indicates how many times a number, called the base, is multiplied by itself, a logarithm tells you the number of times you must multiply the base to get another number. For example, in the expression \( \log_{10} 100 \), this is asking: "10 raised to what power gives us 100?" The answer is 2 because \( 10^2 = 100 \). Thus, \( \log_{10} 100 = 2 \). Logs have specific properties, such as:

• \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \) – the product property.
• \( \log_{b}(\frac{m}{n}) = \log_{b}(m) - \log_{b}(n) \) – the quotient property.
• \( \log_{b}(m^n) = n \cdot \log_{b}(m) \) – the power rule, which will be explained later.

Knowing these can help you manipulate and simplify logarithmic expressions.

Simplifying expressions often requires converting logarithms to a different base, typically base 10 or \( e \), for consistent evaluation or comparison. That's where the change of base formula makes its appearance. The formula is:\[ \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \]This formula allows you to express any logarithm in terms of any other base you choose, which makes calculations easier, especially when using a calculator. For example, let's use the change of base formula to reassess the expression \( \frac{\log_{7} 243}{\log_{7} 3} \). By converting to base 10, we can rewrite it:\[ \frac{\log_{10} 243}{\log_{10} 3} \]Notice how using a common base simplifies calculations. The change of base often reveals patterns or manipulations like cancellations that might not be immediately obvious.

The power rule of logarithms is a convenient tool for simplifying expressions. This rule states that \( \log_{b}(m^n) = n \cdot \log_{b}(m) \). Essentially, it allows you to "bring down" the exponent as a coefficient in front of the log, turning multiplication into an addition problem.This rule is vital when evaluating expressions where arguments of logarithms are powers. For example, consider the logarithmic expression \( \log_{10}(3^5) \). Using the power rule, this can be simplified to:\[ 5 \cdot \log_{10}(3) \]In our earlier solution, that's exactly how we simplified \( \log_{10} 243 \) to find the result \( \frac{5 \log_{10} 3}{\log_{10} 3} = 5 \).Applying the power rule transforms complex log expressions into more approachable arithmetic.