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What does it mean to have a linear relationship but with logarithms? You have a scatter plot, and you can clearly see the relationship between your two variables is not linear. I will try to explain it because teaching is the best way of learning.
How logarithms are used to discover a proportional relationship between two variables ( discover the best line fit)?
There is often no linear relationship between two variables, but we can try another way to see this relationship. But we insist on linearity?
Because the linear relationship is simple. For example, sometimes, the variable x may proportional to y^2 .
Because the linear relationship is simple. For example, sometimes, the variable x may proportional to y^2 .
In this case, we cannot simply find a line that fits our data points. However, we may still be able to fit the data points using a linear relationship using logarithms.
Suppose y=ax^m ; we want to find the value of a and the value of m. If we take the logarithm of both sides: ln(y)= ln(ax^m), now using that neat property of logarithms that say ln(x.y)=ln(x)+ln(y).
so our equation becomes ln(y)= ln(a)+ln(x^m) . And we have another neat algorithm rule that says: ln(x^y)=yln(x). Applying this rule will modify our equation to ln(y)= ln(a)+m ln(x).
This last equation looks like the straight-line equation (y=ax+c), but wrapped up with logarithms. Now we can fit our data using this logarithmic wrapped equation and still get a linear relationship
Please correct any wrong thing you see in this thread
