math 114 test 3 week 8
whenever ____ is the input of log what does it =? ex log10 (1)
0
b^0 =1
1 ex) b^0=1 logb^(x)=y<----> b^y=x =logb^1 +0 y=0, x=1, b=b
recipe for log equations
1) Restrictions 2) Logs on 1 side/ everything else on the other 3) Condense the logs 4) translate (get rid of logs) 5) solve 6) check
expand log4(16 square root of xy)
1) use product prop = log4 (16)+log4( square root of xy) =log4(4)^2 + log4 (xy)^1/2 = 2 +1/2log4(xy) =2+1/2[log4(x) +log4 (y)] = 2+1/2log4(x)+1/2log4(y) ----> final answer
inverse of the common log : log10(x)
1og(x)
convert lnx^2
2lnx
logb (x) = y
<----> b^y=x
how to convert lnx(2)
= ln(x*x)= ln(x) + ln(x) = 1 ln+1 ln =2ln(x)---> FINAL ANSWER
log (1/x)
=log10 (1) -log10 (x) = 0 - log(x) = -log(0)
what is the base of logE(x) or base of the natural log
E
how do you convert this? Log7(10x)
Log7(10)+log7(x)
x = about 0.549 is what type of answer
approximate answer of ask to round/ word problems
b^1
b logb(b)=1
inverse of logb(x)
b(x)
expo form
b^y=x b=base y=expo
e^3=log(y)
ex) 3=logE(y) =3logE(y) =3ln(y)
x=e^-3/5 is what type of answer
exact/ answer to solve
something always has to be _____ the log
inside
inverse of the natural log: logE (x)
ln
convert log2 (10/37)
log2 (10) -log2 (37)
How do you convert? Log6(54)
log6 (9*6) = log6 (9*6) = log6 (9) +log6 (6) = log6 (9)+ 1 ----> FINAL ANSWER
log8 (3x^2)
log8(3x^2)= log8(3) +log8(x^2) =log8(3)+2log8(x) -----> final answer
Quotient property convert
logb (M/N) = logb (M)- logb (N)
Product Property
logb (m*n) = logb (m) + logb (n)
power property
logb(M)^p=p *logb(M)
Change-of-Base Formula for Logarithms
logb(x)= logx/logb ex) log5(87) = log87/log5 = about 2.775 bc round 3 decimal places (final answer)
log(1/x)
logx^-1 =(-1)logx =-logx ----> Final answer because x^-1 = 1/x^1 = 1/x
to condense logs what must you do?
reduce to one log w a coeff of one ex) log8(5)+log8(x) = log8(5x) ex)log6(12)-log6(2)= log6(12/2) = log6(6) = 1 ----> final answer
convert log3( square root of x)
square root of x = x^1/2 =1/2log3 (x)
b^x=b^x
x---->x
log form
y=logbx y=expo b=base
