Evaluate the expression without using a calculator. [tex]$\log _7 1$[/tex]
Answer (1)
Recognize that the logarithm lo g 7 1 asks: to what power must we raise 7 to get 1?
Recall that any non-zero number raised to the power of 0 equals 1.
Apply the rule to find that 7 0 = 1 .
Conclude that lo g 7 1 = 0 .
Explanation
Understanding the problem We are asked to evaluate the expression lo g 7 1 without a calculator. This expression represents the power to which we must raise 7 to obtain 1.
Recalling the definition of logarithm Recall the definition of a logarithm: lo g b a = x means that b x = a . In our case, we have b = 7 and a = 1 . We want to find x such that 7 x = 1 .
Applying the zero exponent rule We know that any non-zero number raised to the power of 0 is equal to 1. Therefore, 7 0 = 1 .
Final Answer Thus, lo g 7 1 = 0 .
Examples
Logarithms are used in many real-world applications, such as measuring the magnitude of earthquakes on the Richter scale. The formula for the Richter scale involves logarithms, allowing us to quantify the energy released by an earthquake. Similarly, logarithms are used in chemistry to measure the acidity or alkalinity of a solution using the pH scale. Understanding logarithms helps us interpret these measurements and make informed decisions based on them.
