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x f(x)
1 409.4
2 260.7
3 170.4
4 110.6
5 74
6 44.7
7 32.4
8 19.5
9 12.7
10 8.1
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x f(x)
0.15 36.21
0.25 28.88
0.5 24.39
0.75 18.28
1 16.5
1.5 12.99
2 9.91
2.25 8.57
2.75 7.23
3 5.99
3.5 4.81

logarithmic; y = 16.68718 9.71860 ln ( x )

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x f(x)
0 9
2 22.6
4 44.2
5 62.1
7 96.9
8 113.4
10 133.4
11 137.6
15 148.4
17 149.3
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Practice test

The population of a pod of bottlenose dolphins is modeled by the function A ( t ) = 8 ( 1.17 ) t , where t is given in years. To the nearest whole number, what will the pod population be after 3 years?

About 13 dolphins.

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Find an exponential equation that passes through the points (0, 4) and (2, 9) .

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Drew wants to save $2,500 to go to the next World Cup. To the nearest dollar, how much will he need to invest in an account now with 6.25 % APR, compounding daily, in order to reach his goal in 4 years?

$ 1,947

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An investment account was opened with an initial deposit of $9,600 and earns 7.4 % interest, compounded continuously. How much will the account be worth after 15 years?

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Graph the function f ( x ) = 5 ( 0.5 ) x and its reflection across the y -axis on the same axes, and give the y -intercept.

y -intercept: ( 0 ,  5 )

Graph of f(-x)=5(0.5)^-x in blue and f(x)=5(0.5)^x in orange.
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The graph shows transformations of the graph of f ( x ) = ( 1 2 ) x . What is the equation for the transformation?

Graph of f(x)= (1/2)^x.
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Rewrite log 8.5 ( 614.125 ) = a as an equivalent exponential equation.

8.5 a = 614.125

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Rewrite e 1 2 = m as an equivalent logarithmic equation.

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Solve for x by converting the logarithmic equation l o g 1 7 ( x ) = 2 to exponential form.

x = ( 1 7 ) 2 = 1 49

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Evaluate log ( 10,000,000 ) without using a calculator.

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Evaluate ln ( 0.716 ) using a calculator. Round to the nearest thousandth.

ln ( 0.716 ) 0.334

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Graph the function g ( x ) = log ( 12 6 x ) + 3.

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State the domain, vertical asymptote, and end behavior of the function f ( x ) = log 5 ( 39 13 x ) + 7.

Domain: x < 3 ; Vertical asymptote: x = 3 ; End behavior: x 3 , f ( x ) and x , f ( x )

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Rewrite log ( 17 a 2 b ) as a sum.

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Rewrite log t ( 96 ) log t ( 8 ) in compact form.

log t ( 12 )

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Rewrite log 8 ( a 1 b ) as a product.

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Use properties of logarithm to expand ln ( y 3 z 2 x 4 3 ) .

3 ln ( y ) + 2 ln ( z ) + ln ( x 4 ) 3

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Condense the expression 4 ln ( c ) + ln ( d ) + ln ( a ) 3 + ln ( b + 3 ) 3 to a single logarithm.

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Rewrite 16 3 x 5 = 1000 as a logarithm. Then apply the change of base formula to solve for x using the natural log. Round to the nearest thousandth.

x = ln ( 1000 ) ln ( 16 ) + 5 3 2.497

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Solve ( 1 81 ) x 1 243 = ( 1 9 ) 3 x 1 by rewriting each side with a common base.

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Use logarithms to find the exact solution for 9 e 10 a 8 5 = 41 . If there is no solution, write no solution .

a = ln ( 4 ) + 8 10

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Find the exact solution for 10 e 4 x + 2 + 5 = 56. If there is no solution, write no solution .

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Find the exact solution for 5 e 4 x 1 4 = 64. If there is no solution, write no solution .

no solution

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Find the exact solution for 2 x 3 = 6 2 x 1 . If there is no solution, write no solution .

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Find the exact solution for e 2 x e x 72 = 0. If there is no solution, write no solution .

x = ln ( 9 )

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Use the definition of a logarithm to find the exact solution for 4 log ( 2 n ) 7 = 11

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Use the one-to-one property of logarithms to find an exact solution for log ( 4 x 2 10 ) + log ( 3 ) = log ( 51 ) If there is no solution, write no solution .

x = ± 3 3 2

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The formula for measuring sound intensity in decibels D is defined by the equation D = 10 log ( I I 0 ) , where I is the intensity of the sound in watts per square meter and I 0 = 10 12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of 4.7 10 1 watts per square meter?

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A radiation safety officer is working with 112 grams of a radioactive substance. After 17 days, the sample has decayed to 80 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance?

f ( t ) = 112 e .019792 t ; half-life: about 35 days

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Write the formula found in the previous exercise as an equivalent equation with base e . Express the exponent to five significant digits.

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A bottle of soda with a temperature of 71° Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of 35° F . After ten minutes, the internal temperature of the soda was 63° F . Use Newton’s Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour?

T ( t ) = 36 e 0.025131 t + 35 ; T ( 60 ) 43 o F

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The population of a wildlife habitat is modeled by the equation P ( t ) = 360 1 + 6.2 e 0.35 t , where t is given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity?

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Enter the data from [link] into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

x f(x)
1 3
2 8.55
3 11.79
4 14.09
5 15.88
6 17.33
7 18.57
8 19.64
9 20.58
10 21.42

logarithmic

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The population of a lake of fish is modeled by the logistic equation P ( t ) = 16 , 120 1 + 25 e 0.75 t , where t is time in years. To the nearest hundredth, how many years will it take the lake to reach 80 % of its carrying capacity?

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For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

x f(x)
1 20
2 21.6
3 29.2
4 36.4
5 46.6
6 55.7
7 72.6
8 87.1
9 107.2
10 138.1

exponential; y = 15.10062 ( 1.24621 ) x

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x f(x)
3 13.98
4 17.84
5 20.01
6 22.7
7 24.1
8 26.15
9 27.37
10 28.38
11 29.97
12 31.07
13 31.43
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x f(x)
0 2.2
0.5 2.9
1 3.9
1.5 4.8
2 6.4
3 9.3
4 12.3
5 15
6 16.2
7 17.3
8 17.9

logistic; y = 18.41659 1 + 7.54644 e 0.68375 x

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Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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