Find the logarithmic equation that fits the following data/table

Question

Below are the corresponding x and y values for the table

X1 = 100, X2 = 50, X3 = 25, X4 = 12.5
Y1 = 0, Y2 = 13, Y3 = 26, Y4 = 39

I clearly see that this data has the multiply-add property and is logarithmic. In fact, I found two different equations that fit this data. The two equations I got are below

Equation #1: a + b*log(x) = 86.37013 – 43.1851 log(x)

Equation #2: log_0.948078(x/100)

Note that the 0.948078 in equation 2 is the log base (rounded). Also, both of the constants in equation 1 are rounded answers as well.
Both equations fit the data perfectly. I obtained the first equation by using logarithmic form a + b*log(x) and solving a system of equations using two of the (x,y) points provided. The second equation I got by interchanging the x and y values to make the equation exponential (since exponential equations are the inverse of logarithmic). Essentially, I found the inverse of the data and then solved for the exponential equation form ab^x . This gave me a = 100 and b = 0.948078. I then found the inverse of this equation which led to logarithmic equation #2 above. Again both equations fit the data perfectly.

You may be thinking, “Well what do you need help with if you have two different workable answers?”. My problem is that the solutions manual to my book gives an answer below

Equation #3: -13*log₂(x/100)

I tested this out in Excel and it also models the data perfectly. How did they get this answer? No steps were provided in the solution. This answer looks much cleaner than my two answers (no rounded numbers) and I’m curious as to where it came from. Please help.

Answer 1

Jeremy, technically both of your answers are correct. In order to get the same answer that you see in the solutions manual, you have to recognize a few things about the multiply-add property of logarithms and how it can relate to the logarithmic equation of a function.

 

The identity I'll be using to get to his answer is below

 

aLog_b(x/c) = -aLog_1/b(x/c)

 

a = the common difference between the y values

b = the common ratio between the x values

c = the x value when y = 0

 

Find the common ratio by dividing any of the x values by the preceding term in your table. You should get 1/2 as the common ratio

Find the common difference by subtracting any of the y values by the preceding term in your table. You should get 13.

These two values probably look familiar to you because you already recognized them when you noticed the multiply-add property of the data.

Using your data, it looks like x is 100 when y = 0.

 

You can now plug these numbers into the identity I used earlier:

 

13Log_1/2(x/100) = -13Log₂(x/100)

 

This is the answer your solutions manual got. I hope this helps.