) is always on the graph of the log function. ( ) The log function is ever increasing, i.e., as we move from left to right the graph rises above. ( ) For x very near to zero, the value of log x can be made lesser than any given real number. In other words in the fourth quadrant the graph approaches y -axis (but never meets it).
( ) Fig . gives the plot of y = e x and y = ln x . It is of interest to observe that the two curves are the mirror images of each other reflected in the line y = x . Two properties of ‘log’ functions are proved below: ( ) There is a standard change of base rule to obtain log a p in terms of log b p .
Let log a p = α , log b p = β and log b a = γ . This means a α = p , b β = p and b γ = a . Substituting the third equation in the first one, we have ( b γ ) α = b γα = p Using this in the second equation, we get b β = p = b γα which implies β = αγ or α = β γ . But then log a p = log b b p ( ) Another interesting property of the log function is its effect on products.
Let log b pq = α . Then b α = pq . If log b p = β and log b q = γ , then b β = p and b γ = q . But then b α = pq = b β b γ = b β + γ which implies α = β + γ , i.e., l og b pq = log b p + log b q A particularly interesting and important consequence of this is when p = q .
In this case the above may be rewritten as log b p = log b p + log