Simplify:
A) log(x+1)+3logx
B) 2log4np-3log4p
C) 2ln(3x)+ln(5x)-ln6xy
D) 22log10+log1000-log1
D) 2log10+log1000-log1
A)
log(x+1)+3logx
log(x+1)+log(x^3)
log (x+1)(x^3)
log(x^4+x^3)
B)
2log(np)-3log(p)
log((np)^2)-log(p^3)
log(n^2p^2)-log(p^3)
log(n^2p^2/p^3)
log(n^2/p)
the base 4 is not important. These steps hold regardless of the base.
If the base is not 4, but the 4 is included in the expression, then the final log is
log(n^2/4p)
C)
2ln(3x)+ln(5x)-ln(6xy)
ln(9x^2*5x/6xy)
ln(15x^2 / 2y)
D)
2log10+log1000-log1
log100+log1000-log1
2+3-0
5
To simplify each of the given expressions, we will use the properties of logarithms and basic algebraic manipulations. Let's go through each expression step by step.
A) log(x+1) + 3log(x)
We can use the property of logarithms that states: log(a) + log(b) = log(ab).
So, applying this property, we can simplify the expression as follows:
log(x+1) + log(x^3)
log(x(x+1))
B) 2log(4np) - 3log(4p)
Using the property log(a) - log(b) = log(a/b), we can rewrite the expression:
log((4np)^2) - log((4p)^3)
log(16n^2p^2) - log(64p^3)
log((16n^2p^2)/(64p^3))
log((n^2)/(4p))
C) 2ln(3x) + ln(5x) - ln(6xy)
Using the property ln(a) + ln(b) - ln(c) = ln((a*b)/c), we can simplify the expression:
ln((3x)^2) + ln(5x) - ln(6xy)
ln(9x^2) + ln(5x) - ln(6xy)
ln((9x^2 * 5x)/(6xy))
ln((45x^3)/(6xy))
ln(45x^2/6y)
ln(15x^2/2y)
D) 22log(10) + log(1000) - log(1)
Using the property log(a) = log(b^c) = c*log(b), we can simplify the expression:
log(10^22) + log(1000) - log(1)
log(10^22) + log(10^3) - log(1)
22log(10) + 3log(10) - 0
22 + 3
25
Therefore, the simplified expressions are:
A) log(x(x+1))
B) log((n^2)/(4p))
C) ln(15x^2/2y)
D) 25