Question 617942: Please help express as a single logarithm.
(loga q - loga r) + 4loga p
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! since they're all log to the base of a, i'll just express them as log with the base of a being implied.
your expression becomes:
log(q) - log(r) + 4*log(p)
take care of the 4*log(p) first.
4*log(p) = log(p^4)
your expression now becomes:
log(q) - log(r) + log(p^4)
this is equivalent to:
log(q/r) + log(p^4) which is equivalent to:
log(q*p^4/r) *****
working your way back out, you get:
log(q*p^4/r) = log(q*p^4) - log(r) which then becomes:
log(q) + log(p^4) - log(r) which then becomes:
log(q) + 4*log(p) - log(r) which then can be arranged to become:
log(q) - log(r) + 4*log(p)
since all of these are to the base of a, then this expression is equivalent to:
log(a,q) - log(a,r) + 4*log(a,p)
the only way to really see if you did it correctly is to assume some values for a, q, r, and p and solve using your calculator.
assume a is equal to 10 which allows you to use your calculator.
assume q is equal to 5
assume r is equal to 10
assume p is equal to 17
your original equation is:
log(a,q) - log(a,r) + 4*log(a,p)
replacing with values given, this becomes:
log(10,5) - log(10,10) + 4*log(10,17)
since log to the base of 10 is assumed by the calculator LOG function, this expression becomes equivalent to:
LOG(5) - LOG(10) + 4*LOG(17)
that expression is equal to 4.62076569
your final expression is:
log(a,(q*p^4/r)
replacing variable names with assigned values, that becomes:
log(10,(5*17^4/10) which becomes:
LOG(5*17^4/10) which becomes:
LOG(41760.5 which is equal to 4.62076569
the answers from the original expression and the final expression are the same so you can assume that you did the translation correctly.
the properties of logarithms that were used are:
log(a*b) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(a^b) = b*log(a)
the reverse of these properties is also true.
log(a) + log(b) = log(a*b)
log(a) - log(b) = log(a/b)
b*log(a) = log(a^b)
the order in which you do the translations is also important.
going from the single form to the multiple form, you needed to resolve the multiplies and divides first.
that's why log(a*p^4/r) was resolved to log(a) + log(p^4) - log(r) which then became log(a) + 4*log(p) - log(r)
going from multiple form to single form, you needed to resolve 4*log(p) first.
that's why log(a) + 4*log(p) - log(r) was resolved to log(a) + log(p^4) - log(r) which then became log(a*p^4/r)
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