SOLUTION: Express in terms of sums and differences of logarithms {{{log((5a/(4b^2)))}}} Write in simplest radical form: {{{root(3,5x^5)}}}

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Express in terms of sums and differences of logarithms {{{log((5a/(4b^2)))}}} Write in simplest radical form: {{{root(3,5x^5)}}}      Log On


   

Question 145816: Express in terms of sums and differences of logarithms
log%28%285a%2F%284b%5E2%29%29%29
Write in simplest radical form:
root%283%2C5x%5E5%29
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Express in terms of sums and differences of logarithms
log(5a/4b^2) = log(5a) - log(4b^2)
= log5 + loga - log4 - 2logb
--------------------------

1n small 3 on top of sqrt sign 5x^5
(5x^5)^(1/3)
= x*(5x^2)
==============
Cheers,
Stan H.

Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
Express in terms of sums and differences of logarithms
log%28%285a%2F%284b%5E2%29%29%29 

Use this rule:----------- log%28%28X%2FY%29%29=log%28%28X%29%29-log%28%28Y%29%29

log%28%285a%29%29-log%28%284b%5E2%29%29

Use this rule:-----------    log%28%28XY%29%29=log%28%28X%29%29%2Blog%28%28Y%29%29
on both terms above being very careful to put the
results of the second term in parentheses because
it has a minus sign in front of it, which will 
change the signs:

%28log%28%285%29%29%2Blog%28%28a%29%29%29-%28log%28%284%29%29%2Blog%28%28b%5E2%29%29%29
 
Now remove the prentheses:

log%28%285%29%29%2Blog%28%28a%29%29-log%28%284%29%29-log%28%28b%5E2%29%29

Use this rule:----------- log%28%28X%5EN%29%29=N%2Alog%28%28X%29%29
on the last term:

log%28%285%29%29%2Blog%28%28a%29%29-log%28%284%29%29-2%2Alog%28%28b%29%29

-----------------------------------------------------

Write in simplest radical form:
root%283%2C5x%5E5%29

Write that as

root%283%2C5%2Ax%2Ax%2Ax%2Ax%2Ax%29

The index is 3 (cube root),
we make as many groups of three like factors
as we can and leave the others ungrouped, like
this:

root%283%2C5%2A%28x%2Ax%2Ax%29%2Ax%2Ax%29

We were only able to make one group in this
problem but in other problems you can make more
groups.

The TRIPLE %28x%2Ax%2Ax%29 comes out in front as a 
SINGLE x and the other factors stay underneath
the radical:

x%2Aroot%283%2C5%2Ax%2Ax%29

Then you write the {{(x*x}}} under the radical as x%5E2

x%2Aroot%283%2C5x%5E2%29

------------------------------
Edwin