Which logarithmic equation is equivalent to $2^5=32$?
A. $\log _2 32=5$
B. $\log _5 32=2$
C. $\log _{32} 5=2$
D. $\log _2 5=32$
Answer (2)
Recognize the exponential form b x = y .
Convert it to logarithmic form lo g b y = x .
Identify base, exponent, and result: 2 5 = 32 .
The equivalent logarithmic equation is lo g 2 32 = 5 .
Explanation
Understanding the Problem We are given the exponential equation 2 5 = 32 and need to find the equivalent logarithmic equation from the given options.
Recalling the Relationship The general form of an exponential equation is b x = y , which is equivalent to the logarithmic equation lo g b y = x , where b is the base, x is the exponent, and y is the result.
Converting to Logarithmic Form In the given equation 2 5 = 32 , the base is 2, the exponent is 5, and the result is 32. Substituting these values into the logarithmic form lo g b y = x , we get lo g 2 32 = 5 .
Identifying the Correct Option Comparing the derived logarithmic equation lo g 2 32 = 5 with the given options, we find that it matches the first option.
Final Answer Therefore, the logarithmic equation equivalent to 2 5 = 32 is lo g 2 32 = 5 .
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes using the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. For example, if we know that a bacterial population doubles every hour, we can use logarithms to determine how long it will take for the population to reach a certain size. This concept is crucial in understanding exponential growth and decay phenomena.
The logarithmic equation equivalent to 2 5 = 32 is lo g 2 32 = 5 . Therefore, the correct option is A. This shows the relationship between exponential and logarithmic forms clearly.
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