If \(\log_8 3 = P\) and \(\log_3 5 = Q\), express \(\log_{10} 5\) in terms of P and Q. Your answer should no longer include any logarithms.

amandapaolars
Apr 3, 2018

#3**+2 **

8^p = 3 and 3^q =5. Sub the first into the 2nd

(8^p)^q = 5

**8^pq = 5**,

Log_10 (5) =**8^(pq)**

Guest Apr 3, 2018

edited by
Guest
Apr 3, 2018

#6**+1 **

P =0.5283208.........

Q =1.4649735.........

P*Q =0.773976......., so:

8^0.773976...... = 5, so:

Log_10(5) = Log_10(8^0.773976...). Remove the "Log"

5 = 8^0.773976 =8^PQ

Guest Apr 3, 2018

#7**0 **

can you explain this further with words... I don't get how you got to that conclusion

amandapaolars
Apr 3, 2018

#8**0 **

Are you having rough day today?

8^p = 3 and 3^q =5.

p =Log(3) / Log(8) =0.52832083357371872715124631464927...........etc.

q =Log(5) / Log(3) =1.4649735207179271671970404076786.............etc.

p*q =0.7739760316291207826234398098298..........etc.

(8^p)^q = 5

8^(p*q) = 8^0.7739760316291207826234398098298.........etc

8^0.7739760316291207826234398098298..... = 5. Sub this for 5 in Log_10(5) and you get

**5 = 8^(pq)**

Guest Apr 3, 2018