Write the equation in its equivalent exponential form.
[tex]$5=\log _9 M$[/tex]
What is the equivalent exponential form of the equation?
Answer (2)
Recall the definition of logarithm: lo g b a = c is equivalent to b c = a .
Apply the definition to the given equation 5 = lo g 9 M , where b = 9 , c = 5 , and a = M .
Convert to exponential form: 9 5 = M .
The equivalent exponential form is: 9 5 = M .
Explanation
Understanding the Problem We are given the logarithmic equation 5 = lo g 9 M and asked to convert it to its equivalent exponential form. Let's recall the relationship between logarithms and exponentials.
Recalling the Definition of Logarithm The logarithm equation lo g b a = c is equivalent to the exponential equation b c = a . In our case, we have b = 9 , c = 5 , and a = M .
Converting to Exponential Form Applying this definition to the given equation 5 = lo g 9 M , we can rewrite it in exponential form as 9 5 = M . To find the value of 9 5 , we can perform the calculation: 9 5 = 9 × 9 × 9 × 9 × 9 = 59049 . Therefore, M = 59049 .
Final Answer The equivalent exponential form of the equation 5 = lo g 9 M is 9 5 = M .
Examples
Logarithmic scales are used to represent large ranges of values in a more manageable way. For example, the Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale. An increase of one unit on the Richter scale corresponds to a tenfold increase in the amplitude of the earthquake waves. Similarly, the decibel scale, used to measure sound intensity, is also logarithmic, where an increase of 10 decibels corresponds to a tenfold increase in sound intensity. Understanding the relationship between logarithmic and exponential forms helps in interpreting and working with these scales.
The equivalent exponential form of the equation 5 = lo g 9 M is 9 5 = M . Hence, M equals 59049 . This transformation illustrates the relationship between logarithms and exponentials.
;
