<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

log 8 2 = 1 3

8 1 / 3 = 2

Got questions? Get instant answers now!

ln ( 1 e 3 ) = −3

e −3 = 1 e 3

Got questions? Get instant answers now!

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

4 −2 = 1 16

log 4 ( 1 16 ) = −2

Got questions? Get instant answers now!

4 −3 / 2 = 0.125

log 4 0.125 = 3 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

log 5 125 x y 3

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

Got questions? Get instant answers now!

ln ( 6 e 3 )

3 2 + ln 6

Got questions? Get instant answers now!

For the following exercises, solve the exponential equation exactly.

e 3 x 15 = 0

ln 15 3

Got questions? Get instant answers now!

7 3 x 2 = 11

2 3 + log 11 3 log 7

Got questions? Get instant answers now!

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

log ( 2 x 7 ) = 0

x = 4

Got questions? Get instant answers now!

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

x = 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

ln x + ln ( x 2 ) = ln 4

1 + 5

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

log 0.5 211

( log 211 log 0.5 7.7211 )

Got questions? Get instant answers now!

log 0.2 0.452

( log 0.452 log 0.2 0.4934 )

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

[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

Got questions? Get instant answers now!

[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.

Got questions? Get instant answers now!

[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.

Got questions? Get instant answers now!

[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.

Got questions? Get instant answers now!

Questions & Answers

f(x) =3+8+4
tennesio Reply
d(x)(x)/dx =?
Abdul Reply
scope of a curve
Abraham Reply
check continuty at x=1 when f (x)={x^3 if x <1 -4-x^2 if -1 <and= x <and= 10
Raja Reply
what is the value as sinx
Sudam Reply
f (x)=x3_2x+3,a=3
Bilal Reply
given demand function & cost function. x= 6000 - 30p c= 72000 + 60x . . find the break even price & quantities.
Fiseha Reply
hi guys ....um new here ...integrate my welcome
Asif Reply
An airline sells tickets from Tokyo to Detroit for $1200. There are 500 seats available and a typical flight books 350 seats. For every $10 decrease in price, the airline observes and additional 5 seats sold. (a) What should the fare be to maximize profit? (b) How many passeners would be on board?
Ravendra Reply
I would like to know if there exists a second category of integration by substitution
CHIFUNDO Reply
nth differential cofficient of x×x/(x-1)(x-2)
Abhay Reply
integral of root of sinx cosx
Wedraj Reply
the number of gallons of water in a tank t minutes after the tank has started to drain is Q(t)=200(30-t)^2.how fast is the water running out at the end of 10 minutes?
Purity Reply
why is it that the integral of eudu =eu
Maman Reply
using L hospital rule
Abubakar Reply
Practice Key Terms 7

Get the best Calculus volume 1 course in your pocket!





Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask