10.1 Rational Exponents
Find the value of x
(1/16)^(-3/4)=x
(2^-4)^((-1/4)(3))=x
2^3=x
x=8
Solve
25^(x+2)=5^(3x-3)
5^(2*(x+2))=5^(3x-3)
2x+4=3x-3
x=7
Suppose
f(x)=2x-1" and "g(x)=x^2+4
a)Find f(g(-2))
b)Find g(f(x))
g(-2)=(-2)^2+4=8
f(g(-2))=f(8)=2(8)-1=15
g(f(x))=(2x-1)^2-1=4x^2-4x+1-1=4x^2-4x=4x(x-1)
Simplify
log_4(64)
log_4(4^3)=3
If
log_10(2)=0.301" and " log_10(3)=0.477
Find
log_10(12)
log_10(12)=log_10(3*2*2)=log_10(3)+log_10(2)+log_10(2)
log_10(12)=.477+.301+.301=1.079
Solve
5^(3t)=2
log_5(5^(3t))=log_5(2)
3t=log_5(2)
t=(log_5(2))/3
t=0.1436
The population of a certain colony of bacteria doubles every 5 hours. How long will it take for the population to triple?
N=N_0(2)^(t/d)
d= 5hours
N/N_0=3
3=2^(t/5)
log_2(3)=log_2(2^(t/5))=t/5
t=(log_2(3))/5=0.3170" hours"
Write x in terms of e
lnx^2=8
2lnx=8
lnx=4
e^4=x
Find the value of x
27^x=81
3^((3)(x))=3^4
3x=4
x=4/3
Solve
4^(x-2)=8^(pi+1)div8^(pi-1)
4^(x-2)=8^((pi+1)-(pi-1))=8^2
2^2(x-2)=2^(3*2)
2x-4=6
x=5
Are the functions f and g inverse functions if
f(x)=5/3x+1" and "g(x)=(3(x-1))/5
f^-1(x)=x=5/3y+1
(3(x-1))/5=y=f^-1(x)=g(x)
Write in exponential form
log_16(1/64)=-3/2
16^(-3/2)=1/64
Solve
log_3(x)=log_3(12)+log_3(2)-log_3(6)
log_3(x)=log_3((12*2)/6)
x=4
Solve
10^(5t)=2
log_10(10^(5t))=log_10(2)
5t=log_10(2)
t=(log_10(2))/5=0.0602
If 40mg of a radioactive substance decays to 5mg in 12 min, find the half-life, in minutes of the substance.
N=N_0(1/2)^(t/h)
N=5
N_0=40
t=12" min"
5=40(1/2)^(12/h)
(1/8)=(1/2)^(12/h)
log_(1/2)(1/8)=log_(1/2)(1/2)^(12/h)=12/h
h=12/(log_(1/2)(1/8))=4
Solve
e^(2x-1)=3
ln(e^(2x-1))=ln3
2x-1=ln3
x=(ln3+1)/2=1.0493
Simplify
root(3)(root()(125y^6)
(5^3y^6)^((1/2)(1/3))
(5y^2)^(1/2)
yroot()(5)
Simplify
((2^(1+root()(2)))/(2^(1-root()(2))))^root()(2)
(2^((1+root()(2))-(1-root()(2))))^root()(2)=(2^(2root()(2)))^root()(2)=2^4=16
Find the inverse of f(x)
f(x)=9x-7
f(x)=y=9x-7
f^-1(x)=x=9y-7
(x+7)/9=y=f^-1(x)=g(x)
Write in logarithmic form with base 2
32^(3/5)=8
(2^5)^(3/5)=8
2^3=8
log_2(8)=3
Write z in terms of x and y
log_b(z)=1/3log_b(x)+log_b(y)
log_b(z)=log_b(x^(1/3))+log_b(y)
log_b(z)=log_b(yx^(1/3))
z=yx^(1/3)
Solve
3^(2x)-7*3^x+12=0
z=3^x" and "z^2=3^(2x)
3^(2x)-7*3^x+12=0
z^2-7z+12=0
(z-3)(z-4)=0
z=3, 4
3^x=3,3^x=4
x=1
log_3(3^x)=log_3(4)
x=log_3(4)=1.262
Bank A offers 6% interest compounded monthly. Bank B offers 6.1% compounded quarterly. Which bank pays more interest per year?
A=P(1+(r/n))^(nt)
Bank A r=0.06 and n=12 P=1 t=1
Bank B r=0.061 and n=4 P=1 t=1
"Bank A" A=(1+(.06/12))^(12)=1.0617
"Bank B" A=(1+(.061/4))^(4)=1.0624
Bank B offers more money
Write as a single natural logarithm
1/2-ln7
1/2=lne^(1/2)
lne^(1/2)-ln7=ln((e^(1/2))/7)