Which equation is equivalent to [tex]$\log _x 36=2$[/tex]?
[tex]$2^x=36$[/tex]
[tex]$x^2=36$[/tex]
[tex]$36^x=2$[/tex]
[tex]$x^2=36^2$[/tex]
Answer (1)
Use the definition of logarithm to convert the given equation to exponential form.
The equation lo g x 36 = 2 is equivalent to x 2 = 36 .
Compare the derived equation with the given options.
The correct equation is x 2 = 36 .
Explanation
Understanding the Problem We are given the equation lo g x 36 = 2 and asked to find an equivalent equation from the given options.
Definition of Logarithm Recall the definition of a logarithm: lo g b a = c is equivalent to 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 the given equation lo g x 36 = 2 , we can rewrite it in exponential form. Here, b = x , a = 36 , and c = 2 . Therefore, the equivalent equation is x 2 = 36 .
Identifying the Correct Option Now, we compare the derived equation x 2 = 36 with the given options:
2 x = 36 (Incorrect) x 2 = 36 (Correct) 3 6 x = 2 (Incorrect) x 2 = 3 6 2 (Incorrect)
The correct option is x 2 = 36 .
Examples
Logarithms are used to solve exponential equations, which are common in calculating population growth, compound interest, and radioactive decay. For example, if you want to know how long it will take for an investment to double at a certain interest rate, you would use logarithms to solve for the time variable in the compound interest formula. Understanding logarithms helps in making informed financial decisions and predicting future trends.
