Which is the exponential form of [tex]$\log _b 35=3$[/tex] ?
[tex]$b^{35}=3$[/tex]
[tex]$b^3=35$[/tex]
[tex]$35^3=b$[/tex]
[tex]$3^b=35$[/tex]
Answer (1)
The exponential form of lo g b 35 = 3 is found by using the definition of a logarithm. The base b raised to the power of 3 equals 35. Therefore, the exponential form is: b 3 = 35
Explanation
Understanding the Problem We are given the logarithmic equation lo g b 35 = 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 b 35 = 3 , we identify a = 35 , b = b , and c = 3 . Therefore, the equivalent exponential form is b 3 = 35 .
Final Answer Thus, the exponential form of lo g b 35 = 3 is b 3 = 35 .
Examples
Logarithms and exponential forms are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. For example, if we know the growth rate of a bacteria population and want to find out how long it takes for the population to reach a certain size, we can use logarithms to solve for the time variable in the exponential growth equation. Understanding the relationship between logarithms and exponential forms allows us to solve real-world problems involving exponential growth and decay.
