SOLUTION: Q. Find the value of the expression {{{ 2/ log( 4, (2000)^6 ) }}} + {{{ 3 / log( 5, (2000)^6 ) }}} My attempt (I would be only writing a select few steps, missing the unnec

Algebra ->  Algebra  -> Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Q. Find the value of the expression {{{ 2/ log( 4, (2000)^6 ) }}} + {{{ 3 / log( 5, (2000)^6 ) }}} My attempt (I would be only writing a select few steps, missing the unnec      Log On

Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!
Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!

   

Question 202615: Q. Find the value of the expression
+2%2F+log%28+4%2C+%282000%29%5E6+%29+ + +3+%2F+log%28+5%2C+%282000%29%5E6+%29+

My attempt (I would be only writing a select few steps, missing the unnececarry ones)
+2%2F+log%28+4%2C+%282000%29%5E6+%29+%2B+3+%2F+log%28+5%2C+%282000%29%5E6+%29+
=
+log%28+4%2C+%284%29%5E2+%29%2F+log%28+4%2C+%282000%29%5E6+%29+%2B+log%28+5%2C+%285%29%5E3+%29+%2F+log%28+5%2C%282000%29%5E6+%29+
=
+log%28+2000%5E6%2C+%284%29%5E2+%29+%2B+log%28+2000%5E6%2C%285%29%5E3+%29+
=
+log%28+2000%2C+%284%29%5E2%2A5%5E3+%29%2F6+
=
+log%28+2000%2C+16%2A625+%29%2F6+
=
+log%28+2000%2C+10000+%29%2F6+%29
I'm confused as to how to procede now!
btw. The answer is 1/6......I just don't know how to get there!
Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!
2%2Flog%284%2C%282000%5E6%29%29%2B3%2Flog%285%2C%282000%5E6%29%29 Start with the given expression.


Let x=2000%5E6


2%2Flog%284%2C%28x%29%29%2B3%2Flog%285%2C%28x%29%29 Replace every instance of 2000%5E6 with "x". This isn't necessary, but I find this substitution simplifies things a bit.


2%2F%28log%2810%2C%28x%29%29%2Flog%2810%2C%284%29%29%29%2B3%2F%28log%2810%2C%28x%29%29%2Flog%2810%2C%285%29%29%29 Use the change of base formula.


Note: the change of base formula is log%28b%2C%28x%29%29=log%2810%2C%28x%29%29%2Flog%2810%2C%28b%29%29



2%2A%28log%2810%2C%284%29%29%2Flog%2810%2C%28x%29%29%29%2B3%2A%28log%2810%2C%285%29%29%2Flog%2810%2C%28x%29%29%29 Multiply by the reciprocal of the lower fractions.


%282%2Alog%2810%2C%284%29%29%29%2Flog%2810%2C%28x%29%29%2B%283%2Alog%2810%2C%285%29%29%29%2Flog%2810%2C%28x%29%29 Multiply.


%282%2Alog%2810%2C%284%29%29%2B3%2Alog%2810%2C%285%29%29%29%2Flog%2810%2C%28x%29%29 Combine the fractions.


%28log%2810%2C%284%5E2%29%29%2Blog%2810%2C%285%5E3%29%29%29%2Flog%2810%2C%28x%29%29 Rewrite each log using the identity y%2Alog%28b%2C%28x%29%29=log%28b%2C%28x%5Ey%29%29


%28log%2810%2C%2816%29%29%2Blog%2810%2C%28125%29%29%29%2Flog%2810%2C%28x%29%29 Square 4 to get 16 and cube 5 to get 125. note: 5%5E3=125 (not 625)


%28log%2810%2C%2816%2A125%29%29%29%2Flog%2810%2C%28x%29%29 Combine the logs using the identity log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29=log%28b%2C%28A%2AB%29%29


%28log%2810%2C%282000%29%29%29%2Flog%2810%2C%28x%29%29 Multiply


%28log%2810%2C%282000%29%29%29%2Flog%2810%2C%282000%5E6%29%29 Plug in x=2000%5E6


%28log%2810%2C%282000%29%29%29%2F%286%2Alog%2810%2C%282000%29%29%29 Rewrite the bottom log using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


%28highlight%28log%2810%2C%282000%29%29%29%29%2F%286%2Ahighlight%28log%2810%2C%282000%29%29%29%29 Highlight the common terms.


%28cross%28log%2810%2C%282000%29%29%29%29%2F%286%2Across%28log%2810%2C%282000%29%29%29%29 Cancel out the common terms.


1%2F6 Simplify


So 2%2Flog%284%2C%282000%5E6%29%29%2B3%2Flog%285%2C%282000%5E6%29%29=1%2F6


You can verify this with a calculator.