Hi. I'm having trouble with this logarithmic equation I'm being asked to do without a calculator. The answer is 8 - the only problem I have is getting there. It's the $${\sqrt[{{\mathtt{{\mathtt{3}}}}}]{{\mathtt{16}}}}$$ that's really messing me up. A little help?

Thanks, guys :)

narchitect Jul 20, 2014

#1**+10 **

Let's take this by parts

log3^{2} + log 16^{1/2} = lg 9 + log 4 = log (9*4) = log(36) .... and.....

[log 12 - log 2] = log (12/2) = log (6)

So, the first fraction becomes.....log(36)/log(6) = log(6)^{2} / log(6) = 2log(6)/log(6) = 2

And the numerator of the second fraction is.......

[2 - log 25 + 2log8] = ] 2 - log25 + log 8^{2} ] = [2 - log25 + log64]

Notice, in terms of logs, that 2 can be written as log 100

So we have..

[log100- log25 + log 64] = [ log (100/25) + log64] = [log(4) + log (64)] =

log (4 * 64) = log(256) = log(16)^{2} = 2log(16)

And the denominator is....

log ^{3}√16 = log 16^{ (1/3)} = (1/3) log 16

And the second fraction becomes...

[2log(16)] / [(1/3)log(16)] = 2/(1/3) = 6

So...combining the result of the first fraction and the second fraction, we have

2 + 6 = 8

And there you go.......

CPhill Jul 20, 2014

#1**+10 **

Best Answer

Let's take this by parts

log3^{2} + log 16^{1/2} = lg 9 + log 4 = log (9*4) = log(36) .... and.....

[log 12 - log 2] = log (12/2) = log (6)

So, the first fraction becomes.....log(36)/log(6) = log(6)^{2} / log(6) = 2log(6)/log(6) = 2

And the numerator of the second fraction is.......

[2 - log 25 + 2log8] = ] 2 - log25 + log 8^{2} ] = [2 - log25 + log64]

Notice, in terms of logs, that 2 can be written as log 100

So we have..

[log100- log25 + log 64] = [ log (100/25) + log64] = [log(4) + log (64)] =

log (4 * 64) = log(256) = log(16)^{2} = 2log(16)

And the denominator is....

log ^{3}√16 = log 16^{ (1/3)} = (1/3) log 16

And the second fraction becomes...

[2log(16)] / [(1/3)log(16)] = 2/(1/3) = 6

So...combining the result of the first fraction and the second fraction, we have

2 + 6 = 8

And there you go.......

CPhill Jul 20, 2014