Which is the exponential form of [tex]$\log _8 x=3$[/tex]?
A. [tex]$3^x=8$[/tex]
B. [tex]$8^3=x$[/tex]
C. [tex]$x^3=8$[/tex]
D. [tex]$8^x=3$[/tex]
Answer (1)
The logarithmic equation lo g 8 x = 3 is converted to its equivalent exponential form using the definition of logarithms. The base 8 raised to the power of 3 equals x, so the exponential form is 8 3 = x . The final answer is 8 3 = x .
Explanation
Understanding the Problem We are given the logarithmic equation lo g 8 x = 3 and asked to find its equivalent exponential form.
Recalling the Definition of Logarithm Recall that a logarithm is the inverse operation to exponentiation. The logarithmic equation lo g b a = c is equivalent to the exponential equation b c = a . In other words, the base b raised to the power of c equals a .
Converting to Exponential Form Applying this definition to our equation lo g 8 x = 3 , we identify the base as b = 8 , the argument of the logarithm as a = x , and the result of the logarithm as c = 3 . Therefore, the equivalent exponential form is 8 3 = x .
Identifying the Correct Option Comparing our result 8 3 = x with the given options, we see that it matches the second option.
Examples
Logarithms and exponentials are used in many real-world applications, such as calculating the magnitude of earthquakes using the Richter scale, modeling population growth, and determining the pH of a solution in chemistry. Understanding how to convert between logarithmic and exponential forms is crucial for solving problems in these areas. For example, if we know the magnitude of an earthquake is 6 on the Richter scale, we can use the exponential form to determine the amplitude of the seismic waves.
