Logs and limits: [ log a (x + h) - log a (x) ] / h

[peblez]

Lim . . .[ log a ( x + h ) – log a x ] / h
h-->0

(The 'a' is supposed to be sub scripted.)

I encounter this problem with out a clue on what to do. Can someone outline some steps and explain the logic to this problem? I absolutely do not know where to begin.


Thank you.

 

[galactus]

Is that 'a' supposed to be the base?. Is it:

\(\displaystyle \L\\\lim_{h\to\0}\frac{log_{a}(x+h)-log_{a}(x)}{h}\)

If so, notice the resemblance to the definition of a derivative:

\(\displaystyle \L\\\lim_{h\to\0}\frac{f(x+h)-f(x)}{h}\)

Your limit should be the same as \(\displaystyle \L\\\frac{d}{dx}[log_{a}(x)]=\frac{1}{x}log_{a}e=\frac{1}{xln(a)}\)

Rewrite your limit as:

\(\displaystyle \L\\\lim_{h\to\0}\frac{1}{h}log_{a}\left(\frac{x+h}{x}\right)\)

\(\displaystyle \L\\\lim_{h\to\0}\frac{1}{h}log_{a}\left(1+\frac{h}{x}\right)\)

Now, let v=h/x and note that v approaches 0 as h approaches 0.

 

[grapz]

thank you galactus