Laws of Exponents
What is the value of:
2^3*2^4
128
What is the y-intercept of the function?
f(x) = 2^x
(0,1)
Fully expand the logarithm:
log_b(xy)
log_b(x)+log_b(y)
A city's pigeon population grows every month at a rate of 1.8%. If the current population is 2350, what will the population be after 5 years?
Round to the nearest pigeon.
~6854 pigeons will reside in the city after 5 years
f(t)=2350(1+0.018)^t
t=60
Convert to logarithmic form:
y=2^x
log_2(y)=x
Solve for all real values of x:
log(x + 2) - \log(3) = 1
x=28
Simplify the following:
(3^8)^(1/2
81
What is the domain of the function?
g(x) = \log(x)
All real numbers larger than 0
{x|x>0}
Fully expand the logarithm:
log_a(5x^2)
log_a(5)+2log_a(x)
A population of bacteria decays by 16% every 4 hours and began with 85000 individuals. Find the population after 3 days have passed.
Round to the nearest individual.
~3685 bacteria will be present after 3 days
f(t)=85000(1-0.16)^t
t=18
Convert to exponential form:
y = log_3(x)
3^y=x
Solve for x:
3^{2x} = 27
x=3/2
Given that a=3, find:
a^0
a^0=1
Describe the transformations:
h(x) = 3^{x+1} - 4
Vertical stretch by 3.
Translated Left 1.
Translated Down 4.
Fully condense the logarithm:
3log(a)+log(b)-log(c)
log((a^3b)/c)
A radioactive substance has a half-life of 2 years. If you start with 800 grams, how much will remain after 15 years?
Round to the nearest hundredth of a gram.
~4.42 grams will remain after 15 years
f(t)=800(1-0.5)^(t/2)
t=15
Rewrite in log form:
5^x = 25
log_5(25)=x
Solve for all real values of x:
log_5(x^2 - 4) = 1
x=+-3
Fully simplify:
(2a^7b^5)/(4a^3b^7)
(a^4)/(2b^2)
Write the equation of the asymptote for the following function:
f(x) = 3^{x} - 5
y=-5
Fully condense:
log(10) + 2 \log(3) - \log(2)
log(180)
Solve the exponential growth formula for time:
f(t)=a(1+r)^t
t=ln(f(t)/a)/ln(1+r)
Rewrite in exponential form:
log(x) = 3
10^3=x
Solve for x, no need to simplify:
e^{2x-5} = 7
x=(ln(7)+5)/2
Fully simplify:
(1/2a^3b^5)^2/(a^6b^10)
1/4
What would the coordinates of (1,0) from g(x) become after the transformation into f(x)?
g(x)=log_2(x)
f(x)=log_2(-x-7)+1
(-8,1)
Fully expand the logarithm:
log((x^2y^3)/z^4)
2logx+3logy-4logz
Solve the continuous exponential growth formula for rate:
f(t)=Pe^(rt
r=ln(f(t)/P)/t
Convert to exponential form:
ln(x)=8
e^8=x
Solve for all real values of x:
log_2(x^2-5x+5)=log_2(4x-37)
No real solution!