Given that [tex]$10^4=10,000$[/tex], write this in logarithm form.
Answer (1)
Recognize the exponential form: 1 0 4 = 10 , 000 .
Apply the definition of logarithm: If b x = y , then lo g b y = x .
Convert the exponential equation to logarithmic form: lo g 10 10 , 000 = 4 .
Simplify the logarithmic form: lo g 10 , 000 = 4 .
Explanation
Understanding the Problem We are given the exponential equation 1 0 4 = 10 , 000 and asked to write it in logarithmic form.
Recalling the Definition of Logarithm Recall the definition of a logarithm: If b x = y , then the equivalent logarithmic form is lo g b y = x , where b is the base, x is the exponent, and y is the result.
Identifying the Base, Exponent, and Result In our case, we have 1 0 4 = 10 , 000 . Comparing this to b x = y , we identify b = 10 , x = 4 , and y = 10 , 000 .
Writing in Logarithmic Form Applying the definition, we can write the logarithmic form as lo g 10 10 , 000 = 4 . Since the base is 10, we can simply write it as lo g 10 , 000 = 4 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (Richter scale), the loudness of sound (decibels), and the acidity of a solution (pH scale). Understanding how to convert between exponential and logarithmic forms allows us to work with these scales effectively. For example, if an earthquake measures 7 on the Richter scale, we can use logarithms to determine the amplitude of the seismic waves.
