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f ( x ) = e x + 2

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the line “y = 2” and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 3).

Domain: all real numbers, range: ( 2 , ) , y = 2

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f ( x ) = 3 x + 1

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 3). Another point of the graph is at (-1, 1).

Domain: all real numbers, range: ( 0 , ) , y = 0

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f ( x ) = 1 2 x

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that increases until it comes close the line “y = 1” without touching it. There x intercept and the y intercept are both at the origin. Another point of the graph is at (-1, -1).

Domain: all real numbers, range: ( , 1 ) , y = 1

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f ( x ) = e x 1

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved decreasing function that decreases until it comes close the line “y = -1” without touching it. There x intercept and the y intercept are both at the origin. There is an approximate point on the graph at (-1, 1.7).

Domain: all real numbers, range: ( −1 , ) , y = −1

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For the following exercises, write the equation in equivalent exponential form.

log 8 2 = 1 3

8 1 / 3 = 2

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ln ( 1 e 3 ) = −3

e −3 = 1 e 3

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For the following exercises, write the equation in equivalent logarithmic form.

4 −2 = 1 16

log 4 ( 1 16 ) = −2

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4 −3 / 2 = 0.125

log 4 0.125 = 3 2

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For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote.

f ( x ) = ln ( x 1 )

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line “x = 1”. There is no y intercept and the x intercept is at the approximate point (2, 0).

Domain: ( 1 , ) , range: ( , ) , x = 1

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f ( x ) = 1 ln x

An image of a graph. The x axis runs from -1 to 9 and the y axis runs from -5 to 5. The graph is of a decreasing curved function which starts slightly to the right of the y axis. There is no y intercept and the x intercept is at the point (e, 0).

Domain: ( 0 , ) , range: ( , ) , x = 0

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f ( x ) = ln ( x + 1 )

An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line “x = -1”. There y intercept and the x intercept are both at the origin.

Domain: ( −1 , ) , range: ( , ) , x = −1

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For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.

log 3 9 a 3 b

2 + 3 log 3 a log 3 b

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log 5 125 x y 3

3 2 + 1 2 log 5 x + 3 2 log 5 y

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ln ( 6 e 3 )

3 2 + ln 6

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For the following exercises, solve the exponential equation exactly.

e 3 x 15 = 0

ln 15 3

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7 3 x 2 = 11

2 3 + log 11 3 log 7

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For the following exercises, solve the logarithmic equation exactly, if possible.

log ( 2 x 7 ) = 0

x = 4

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log 6 ( x + 9 ) + log 6 x = 2

x = 3

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log 4 ( x + 2 ) log 4 ( x 1 ) = 0

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ln x + ln ( x 2 ) = ln 4

1 + 5

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For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

log 7 82

( log 82 log 7 2.2646 )

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log 0.5 211

( log 211 log 0.5 7.7211 )

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log 0.2 0.452

( log 0.452 log 0.2 0.4934 )

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Rewrite the following expressions in terms of exponentials and simplify.

a. 2 cosh ( ln x ) b. cosh 4 x + sinh 4 x c. cosh 2 x sinh 2 x d. ln ( cosh x + sinh x ) + ln ( cosh x sinh x )

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[T] The number of bacteria N in a culture after t days can be modeled by the function N ( t ) = 1300 · ( 2 ) t / 4 . Find the number of bacteria present after 15 days.

~ 17 , 491

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[T] The demand D (in millions of barrels) for oil in an oil-rich country is given by the function D ( p ) = 150 · ( 2.7 ) −0.25 p , where p is the price (in dollars) of a barrel of oil. Find the amount of oil demanded (to the nearest million barrels) when the price is between $15 and $20.

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[T] The amount A of a $100,000 investment paying continuously and compounded for t years is given by A ( t ) = 100,000 · e 0.055 t . Find the amount A accumulated in 5 years.

Approximately $131,653 is accumulated in 5 years.

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[T] An investment is compounded monthly, quarterly, or yearly and is given by the function A = P ( 1 + j n ) n t , where A is the value of the investment at time t , P is the initial principle that was invested, j is the annual interest rate, and n is the number of time the interest is compounded per year. Given a yearly interest rate of 3.5% and an initial principle of $100,000, find the amount A accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly.

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Questions & Answers

what is the limit of x^2+x when x approaches 0
Dike Reply
find derivatives 3√x²+√3x²
Care Reply
3 + 3=6
mujahid
How to do basic integrals
dondi Reply
write something lmit
ram Reply
find the integral of tan tanxdx
Lateef Reply
-ln|cosx| + C
Jug
discuss continuity of x-[x] at [ _1 1]
Atshdr Reply
Given that u = tan–¹(y/x), show that d²u/dx² + d²u/dy²=0
Collince Reply
find the limiting value of 5n-3÷2n-7
Joy Reply
Use the first principal to solve the following questions 5x-1
Cecilia Reply
175000/9*100-100+164294/9*100-100*4
Ibrahim Reply
mode of (x+4) is equal to 10..graph it how?
Sunny Reply
66
ram
6
ram
6
Cajab
what is domain in calculus
nelson
integrals of 1/6-6x-5x²
Namwandi Reply
derivative of (-x^3+1)%x^2
Misha Reply
(-x^5+x^2)/100
Sarada
(-5x^4+2x)/100
Sarada
oh sorry it's (-x^3+1)÷x^2
Misha
-5x^4+2x
Sarada
sorry I didn't understan A with that symbol
Sarada
find the derivative of the following y=4^e5x y=Cos^2 y=x^inx , x>0 y= 1+x^2/1-x^2 y=Sin ^2 3x + Cos^2 3x please guys I need answer and solutions
Ga Reply
differentiate y=(3x-2)^2(2x^2+5) and simplify the result
Ga
72x³-72x²+106x-60
okhiria
y= (2x^2+5)(3x+9)^2
lemmor
solve for dy/dx of y= 8x^3+5x^2-x+5
Ga Reply
192x^2+50x-1
Daniel
are you sure? my answer is 24x^2+10x-1 but I'm not sure about my answer .. what do you think?
Ga
24x²+10x-1
Eyad
eyad Amin that's the correct answer?
Ga
yes
Eyad
ok ok hehe thanks nice dp ekko hahaha
Ga
hahaha 😂❤️❤️❤️ welcome bro ❤️
Eyad
eyad please answer my other question for my assignment
Ga
y= (2x^2+5)(3x+9)^2
lemmor
can i join?
Fernando
yes of course
Jug
can anyone teach me integral calculus?
Jug
it's just the opposite of differential calculus
yhin
of coursr
okhiria
but i think, it's more complicated than calculus 1
Jug
Hello can someone help me with calculus one...
Jainaba
Practice Key Terms 7

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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