Which is the logarithmic form of [tex]$6^4=1296$[/tex]?
[tex]$\log 1296=4$[/tex]
[tex]$\log 1296=4 \bullet 6$[/tex]
[tex]$\log _4 1296=6$[/tex]
[tex]$\log _6 1296=4$[/tex]
Answer (1)
The exponential form is 6 4 = 1296 .
Convert the exponential form to logarithmic form using the definition: lo g b y = x if b x = y .
Identify the base, exponent, and result: b = 6 , x = 4 , y = 1296 .
The logarithmic form is lo g 6 1296 = 4 .
Explanation
Understanding the Problem We are given the exponential equation 6 4 = 1296 and asked to find its equivalent logarithmic form.
Recalling the Definition of Logarithm Recall that the exponential form b x = y is equivalent to the logarithmic form 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 6 4 = 1296 , so b = 6 , x = 4 , and y = 1296 . Applying the definition, we get lo g 6 1296 = 4 .
Final Answer Therefore, the logarithmic form of 6 4 = 1296 is lo g 6 1296 = 4 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (Richter scale), the loudness of sound (decibels), and the acidity of a solution (pH scale). They are also used in computer science to analyze the complexity of algorithms and in finance to calculate compound interest. Understanding how to convert between exponential and logarithmic forms is essential for solving problems in these areas.
