logarithms qn?

Question

given that lg x=p and lg y=q, express the following in terms of p and q.

(a) lg [100(square root x)]/(y^2)
ans: 2+ 1/2p-2q
(b) lg (y^x)
ans: q(10^p)

for qn(b)
why cant i get the ans if i solve in this manner

lg(y^x)= xlg y
= x(y)
=xy
why cant i do this?

Answer 1

(a) 10^p = x
10^q = y
y^2 = 10^(2q) = 100^q

log (100 square root x) / y^2
log (100 (square root 10^p)) / 100^q
*If you have a log in which something inside the parenthesis is multiplied, you can separate the logs and add them together*

So:
log (100 (square root x)) = log (100) + log (square root x)
*Also, any exponent can be pulled out from inside the parenthesis and put out as a coefficient infront of the log*

So: log (100) + log (square root x) = log (100) + (1/2) log (x)
log (100) = 2

= (2+ (1/2)log(x)) / (log(y)^2)
*Anytime that you divide by a log function you can change it to subtraction.*

= 2 + (1/2) log (x) - log (y)^2
= 2 + (1/2) log (x) -2log(y)
Now you can change to p's and q's.

= 2 + (1/2) p -2q


(b) *You can take the x out because it is an exponent and put it out front of the log*

x log (y)
also, we know:

x=10^p *because you take the base of the log (which is always 10 unless specified otherwise), raise it to the number or letter that it equals, and set it equal to the number or letter inside the parenthesis in the log function*

so: log(x) = p : 10^p =x

so, you can add in the p's and q's
q(10^p) = log(y^x)

Answer 2

log(y^x) = x*log(y). You are given that q = log(y) you have x*q, but you are also given log(x) = p, which means x = 10^p, so x*q = 10^p * q or q*(10^p)

So: 1) you cannot replace log(y) with y, and you need your answer in terms of p and q.

Answer 3

x log y # xy, rather it equals xq. From the definition of p, x = 10^p.