What are the values of h and k ?

Question

Given that #log_2 hk = 3# and #log_2 h^3 k^2 = 5# , find the values of h and k.

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Answer 1 · 2017-09-03T16:37:12

#h=1/2, k=16#

One approach is as follows

using the laws of logs

#log_ab=c=>a^c=b#

#log_aX+log_aY=log_a(XY)#

#log_aX^n=nlog_aY#

#-----------------#

#log_2(hk)=3=>color(red)(log_2h+log_2k=3--(1))#

#log_2(h^3k^2)=5#

#=>log_2h^3+log_2k^2=5#

#=>color(blue)(3log_2h+2log_2k=5---(2))#

#xx(1) " "by" " 2#

#color(red)(2log_2h+2log_2k=6--(1))#

#color(blue)(3log_2h+2log_2k=5---(2))#

Subtract

#-log_2h=1#

#=>log_2h=-1=>h=1/2#

substitute back into #(1)#

#log_2h+log_2k=3--(1))#

gives

#-1+log_2k=3#

#=>log_2k=4#

#=>k=2^4=16#

so #h=1/2, k=16#

checking the second eqn for consistency

3log_2h+2log_2k=5

#LHS" "3xx-1+2xx4=-3+8=5=RHS#