Given that [tex]$3^2=9$[/tex], write this in logarithm form.
A. [tex]$\log _2 3=9$[/tex]
B. [tex]$\log _3 9=2$[/tex]
C. [tex]$\log _2 9=3$[/tex]
D. [tex]$\log _3 2=9$[/tex]
Answer (1)
The exponential equation 3 2 = 9 is converted to logarithmic form using the definition of a logarithm. The base 3 raised to the power of 2 equals 9, which translates to the logarithm of 9 with base 3 equals 2. Therefore, the logarithmic form is lo g 3 9 = 2 .
Explanation
Understanding the Problem We are given the exponential equation 3 2 = 9 and asked to express it in logarithmic form.
Recalling the Definition of Logarithm Recall that the exponential form b x = y can be written in logarithmic form as lo g b y = x , where b is the base, x is the exponent, and y is the result.
Applying the Definition to the Given Equation In our case, we have 3 2 = 9 , so b = 3 , x = 2 , and y = 9 . Applying the definition, we get lo g 3 9 = 2 .
Identifying the Correct Option Comparing this with the given options, we see that the correct logarithmic form is lo g 3 9 = 2 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sounds (decibels), and the acidity of a solution (pH). Understanding how to convert between exponential and logarithmic forms is crucial for working with these applications. For example, if an earthquake measures 7 on the Richter scale, it means the amplitude of the seismic waves is 1 0 7 times larger than a standard level.
