### By Moni Olubuyide

First let us begin with the parent functions of natural log and log of x.

Notice that both graphs intersect at (1,0).

Now lets looks how the graph of ln x is transformed when it is multiplied by a constant. The graph below shows when the equation is

y = 4 (ln x). Notice that the values of ln x now has a stretch factor of 4.

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What happens when x is multiplied by a constant.....

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Notice now that the values of ln x has shifted up by ln n. For example, ln 2 = .69 and ln (4*2) = 2.08....Well ln 4 = 1.39, so at x =2 the graph shifted up by 1.39. Therefore we can say that ln (nx) = ln n + ln x

Now let's observe what happens with a logarithmic function when it is multiplied by a constant. The graph you see below is

y = 4 (log x)

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Just like the natural log graphs...multiplying by n gives a stretch factor of n. Comparing the graphs log x with 4 (log x), you can see that the values of 4 (log x) are four times the values of log x!!!

Since the logarithmic function above had similar results to the natural log function, I will hypothesis that when I multiply x by a constant I will get log (nx) = log n + log x.....

CLICK HERE FOR ANIMATION to investigate the values

Analyzing the values between both graphs.... is my hypothesis correct??? Yes.. at x =2...log 2 = .301 and log (4*2) = .903 which is the same as log 4 + log 2 since log 4 = .602...