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# log(19, 0.5) + log(19, 2) = 0 how to explain this?

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$${{log}}_{{\mathtt{0.5}}}{\left({\mathtt{19}}\right)}{\mathtt{\,\small\textbf+\,}}{{log}}_{{\mathtt{2}}}{\left({\mathtt{19}}\right)} = {\mathtt{0}}$$

how to explain this?

Guest Jan 6, 2015

### Best Answer

#2
+85819
+10

Here's a slightly different approach to Melody's

Note that we can write .5  as 1/2  = 2-1

So we have

log2(19)  + log0.5 (19)   .....and...using the"change of base" rule, we can write

log(19) / log(2)  +  log(19) / log(.05) =

log(19) / log(2)  + log(19) / log(1/2) =

log(19) / log(2)  +  log(19) / log2-1 =

log(19) / log(2)  +  log(19) / (-1) log2 =

log(19) / log(2)  +  [ (-1) log(19) /  log2 ] =

log(19) / log(2)  -  log(19) /  log2 =   0

CPhill  Jan 7, 2015
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### 3+0 Answers

#1
+92225
+5

Mmm interesting.

Let

$$\\x=log_{0.5}(19)\qquad and \qquad y=log_2(19)\\ so\\ 19=0.5^x\qquad\qquad and \qquad 19=2^y\\ 19=(\frac{1}{2})^x\\ 2^x=\frac{1}{19}\\ so\\ 2^x\times 2^y=\frac{1}{19}\times 19\\ 2^x\times 2^y=1\\ 2^x\times 2^y=2^0\\ 2^{x+y}=2^0\\ therefore\\ x+y=0\\ so\\ log_{0.5}(19)+log_2(19)=0$$

Melody  Jan 7, 2015
#2
+85819
+10
Best Answer

Here's a slightly different approach to Melody's

Note that we can write .5  as 1/2  = 2-1

So we have

log2(19)  + log0.5 (19)   .....and...using the"change of base" rule, we can write

log(19) / log(2)  +  log(19) / log(.05) =

log(19) / log(2)  + log(19) / log(1/2) =

log(19) / log(2)  +  log(19) / log2-1 =

log(19) / log(2)  +  log(19) / (-1) log2 =

log(19) / log(2)  +  [ (-1) log(19) /  log2 ] =

log(19) / log(2)  -  log(19) /  log2 =   0

CPhill  Jan 7, 2015
#3
+92225
0

Thanks Chris

It is always good to get a couple of different takes

Melody  Jan 7, 2015

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