Exam 5: Exponential and Logarithmic Functions

Rewrite the expression as the sum and/or difference of logarithms, without using exponents. Simplify if possible. - $\log _ { n } \sqrt [ 7 ] { \frac { 6 x ^ { 5 } } { z ^ { 9 } } }$

(Multiple Choice)

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Question 81

Match the equation with its graph. - $y = 4 ^ { x + 5 }$

(Multiple Choice)

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Question 82

Provide an appropriate response. -Consider the logistic function $f ( x ) = \frac { c } { 1 + a e ^ { - b x } } , \text { where } b > 0$ 0. Is this function increasing or decreasing?

(Short Answer)

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Question 83

$2 ^ { ( 12 - 3 x ) } = 64$

(Multiple Choice)

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Question 84

The number of years t it takes for an investment to double if it earns r percent (as a decimal), compounded annually, $t = \frac { \ln 2 } { \ln ( 1 + r ) }$ . How long will it take for prices in the economy to double at a 13% annual inflation Rate? Round to the nearest year.

(Multiple Choice)

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Question 85

Write the logarithmic equation in exponential form. - $\log _ { 10 } 0 = - 10$

(Multiple Choice)

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Question 86

Provide an appropriate response. -Consider the logistic function $f ( x ) = \frac { c } { 1 + a e ^ { - b x } } , \text { where } b > 0$ . What is the initial value of this function?

(Essay)

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Question 87

Find the exponential function that satisfies the given conditions. -Initial value $= 55$ , decreasing at a rate of $0.46 \%$ per week

(Multiple Choice)

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Question 88

The number of quarters needed to double an investment when a lump sum is invested at 4%, compounded quarterly, is given b $\mathrm { n } = \frac { \ln 2 } { \ln 1.01 }$ . In how many years will the investment double, to the nearest tenth of a year?

(Multiple Choice)

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Question 89

Explain the error in the following: $\log _ { 3 } 2 + \log _ { 3 } \mathrm { M } = \log _ { 3 } ( 2 + \mathrm { M } )$

(Essay)

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Question 90

Find the value of the logarithm without using a calculator. - $\log _ { 10 } 100$

(Multiple Choice)

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Question 91

$\mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { k } } \right) ^ { \mathrm { kn } } \text { for } \mathrm { P } = \$ 12,000 , \mathrm { n } = 6 , \mathrm { r } = 8 \% , \mathrm { k } = 4$

(Multiple Choice)

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Question 92

The number of periods needed to double an investment when a lump sum is invested at 2%, compounded semiannually, is given by n $\mathrm { n } = \log _ { 1.01 } 2$ 2. Use the change of base formula to find n, rounded to the nearest tenth.

(Multiple Choice)

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Question 93

What will be the amount in an account with initial principal $9000 if interest is compounded continuously at an annual rate of 7.25% for 7 years?

(Multiple Choice)

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Question 94

Provide an appropriate response. -Using A $\mathrm { A } = \mathrm { P } ( 1 + r \mathrm { t } )$ ), the future value formula for a simple interest investment, derive the formula for t, the time (in years).

(Short Answer)

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Question 95

Use a change of base formula to evaluate the given logarithm. Approximate to three decimal places. - $\log _ { 8 } ( 0.662 )$

(Multiple Choice)

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Question 96

The number of periods needed to double an investment when a lump sum is invested at 11%, compounded semiannually, is given by n $\mathrm { n } = \frac { \ln 2 } { \ln 1.055 }$ . Find n, rounded to the nearest tenth.

(Multiple Choice)

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Question 97

$\mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { k } } \right) ^ { \mathrm { kn } } \text { for } \mathrm { P } = \$ 480 , \mathrm { n } = 3 , \mathrm { r } = 5 \% , \mathrm { k } = 4$

(Multiple Choice)

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Question 98

The number of quarters needed to double an investment when a lump sum is invested at 10%, compounded quarterly, is given b $\mathrm { n } = \log _ { 1.025 } 2$ In how many years will the investment double, to the nearest tenth of a Year?

(Multiple Choice)

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Question 99

Provide an appropriate response. -Consider the Gompertz function $\mathrm { N } = \mathrm { Ca } \mathrm { R } ^ { \mathrm { t } } , \text { where } 0 < \mathrm { R } < 1$ . What is the initial value of this function?

(Short Answer)

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Question 100

Rewrite the expression as the sum and/or difference of logarithms, without using exponents. Simplify if possible. - $\log _ { n } \sqrt [ 7 ] { \frac { 6 x ^ { 5 } } { z ^ { 9 } } }$

Match the equation with its graph. - $y = 4 ^ { x + 5 }$

Provide an appropriate response. -Consider the logistic function $f ( x ) = \frac { c } { 1 + a e ^ { - b x } } , \text { where } b > 0$ 0. Is this function increasing or decreasing?

$2 ^ { ( 12 - 3 x ) } = 64$

The number of years t it takes for an investment to double if it earns r percent (as a decimal), compounded annually, $t = \frac { \ln 2 } { \ln ( 1 + r ) }$ . How long will it take for prices in the economy to double at a 13% annual inflation Rate? Round to the nearest year.

Write the logarithmic equation in exponential form. - $\log _ { 10 } 0 = - 10$

Provide an appropriate response. -Consider the logistic function $f ( x ) = \frac { c } { 1 + a e ^ { - b x } } , \text { where } b > 0$ . What is the initial value of this function?

Find the exponential function that satisfies the given conditions. -Initial value $= 55$ , decreasing at a rate of $0.46 \%$ per week

The number of quarters needed to double an investment when a lump sum is invested at 4%, compounded quarterly, is given b $\mathrm { n } = \frac { \ln 2 } { \ln 1.01 }$ . In how many years will the investment double, to the nearest tenth of a year?

Explain the error in the following: $\log _ { 3 } 2 + \log _ { 3 } \mathrm { M } = \log _ { 3 } ( 2 + \mathrm { M } )$

Find the value of the logarithm without using a calculator. - $\log _ { 10 } 100$

$\mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { k } } \right) ^ { \mathrm { kn } } \text { for } \mathrm { P } = \$ 12,000 , \mathrm { n } = 6 , \mathrm { r } = 8 \% , \mathrm { k } = 4$

The number of periods needed to double an investment when a lump sum is invested at 2%, compounded semiannually, is given by n $\mathrm { n } = \log _ { 1.01 } 2$ 2. Use the change of base formula to find n, rounded to the nearest tenth.

What will be the amount in an account with initial principal $9000 if interest is compounded continuously at an annual rate of 7.25% for 7 years?

Provide an appropriate response. -Using A $\mathrm { A } = \mathrm { P } ( 1 + r \mathrm { t } )$ ), the future value formula for a simple interest investment, derive the formula for t, the time (in years).

Use a change of base formula to evaluate the given logarithm. Approximate to three decimal places. - $\log _ { 8 } ( 0.662 )$

The number of periods needed to double an investment when a lump sum is invested at 11%, compounded semiannually, is given by n $\mathrm { n } = \frac { \ln 2 } { \ln 1.055 }$ . Find n, rounded to the nearest tenth.

$\mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { k } } \right) ^ { \mathrm { kn } } \text { for } \mathrm { P } = \$ 480 , \mathrm { n } = 3 , \mathrm { r } = 5 \% , \mathrm { k } = 4$

The number of quarters needed to double an investment when a lump sum is invested at 10%, compounded quarterly, is given b $\mathrm { n } = \log _ { 1.025 } 2$ In how many years will the investment double, to the nearest tenth of a Year?

Provide an appropriate response. -Consider the Gompertz function $\mathrm { N } = \mathrm { Ca } \mathrm { R } ^ { \mathrm { t } } , \text { where } 0 < \mathrm { R } < 1$ . What is the initial value of this function?

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Exam 5: Exponential and Logarithmic Functions

Rewrite the expression as the sum and/or difference of logarithms, without using exponents. Simplify if possible. - log⁡n6x5z97\log _ { n } \sqrt [ 7 ] { \frac { 6 x ^ { 5 } } { z ^ { 9 } } }logn​7z96x5​​

Match the equation with its graph. - y=4x+5y = 4 ^ { x + 5 }y=4x+5

Provide an appropriate response. -Consider the logistic function f(x)=c1+ae−bx, where b>0f ( x ) = \frac { c } { 1 + a e ^ { - b x } } , \text { where } b > 0f(x)=1+ae−bxc​, where b>0 0. Is this function increasing or decreasing?

2(12−3x)=642 ^ { ( 12 - 3 x ) } = 642(12−3x)=64

Write the logarithmic equation in exponential form. - log⁡100=−10\log _ { 10 } 0 = - 10log10​0=−10

Provide an appropriate response. -Consider the logistic function f(x)=c1+ae−bx, where b>0f ( x ) = \frac { c } { 1 + a e ^ { - b x } } , \text { where } b > 0f(x)=1+ae−bxc​, where b>0 . What is the initial value of this function?

Find the exponential function that satisfies the given conditions. -Initial value =55= 55=55 , decreasing at a rate of 0.46%0.46 \%0.46% per week

The number of quarters needed to double an investment when a lump sum is invested at 4%, compounded quarterly, is given b n=ln⁡2ln⁡1.01\mathrm { n } = \frac { \ln 2 } { \ln 1.01 }n=ln1.01ln2​ . In how many years will the investment double, to the nearest tenth of a year?

Explain the error in the following: log⁡32+log⁡3M=log⁡3(2+M)\log _ { 3 } 2 + \log _ { 3 } \mathrm { M } = \log _ { 3 } ( 2 + \mathrm { M } )log3​2+log3​M=log3​(2+M)

Find the value of the logarithm without using a calculator. - log⁡10100\log _ { 10 } 100log10​100

P(1+rk)kn for P=$12,000,n=6,r=8%,k=4\mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { k } } \right) ^ { \mathrm { kn } } \text { for } \mathrm { P } = \$ 12,000 , \mathrm { n } = 6 , \mathrm { r } = 8 \% , \mathrm { k } = 4P(1+kr​)kn for P=$12,000,n=6,r=8%,k=4

The number of periods needed to double an investment when a lump sum is invested at 2%, compounded semiannually, is given by n n=log⁡1.012\mathrm { n } = \log _ { 1.01 } 2n=log1.01​2 2. Use the change of base formula to find n, rounded to the nearest tenth.

What will be the amount in an account with initial principal $9000 if interest is compounded continuously at an annual rate of 7.25% for 7 years?

Provide an appropriate response. -Using A A=P(1+rt)\mathrm { A } = \mathrm { P } ( 1 + r \mathrm { t } )A=P(1+rt) ), the future value formula for a simple interest investment, derive the formula for t, the time (in years).

Use a change of base formula to evaluate the given logarithm. Approximate to three decimal places. - log⁡8(0.662)\log _ { 8 } ( 0.662 )log8​(0.662)

The number of periods needed to double an investment when a lump sum is invested at 11%, compounded semiannually, is given by n n=ln⁡2ln⁡1.055\mathrm { n } = \frac { \ln 2 } { \ln 1.055 }n=ln1.055ln2​ . Find n, rounded to the nearest tenth.

P(1+rk)kn for P=$480,n=3,r=5%,k=4\mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { k } } \right) ^ { \mathrm { kn } } \text { for } \mathrm { P } = \$ 480 , \mathrm { n } = 3 , \mathrm { r } = 5 \% , \mathrm { k } = 4P(1+kr​)kn for P=$480,n=3,r=5%,k=4

The number of quarters needed to double an investment when a lump sum is invested at 10%, compounded quarterly, is given b n=log⁡1.0252\mathrm { n } = \log _ { 1.025 } 2n=log1.025​2 In how many years will the investment double, to the nearest tenth of a Year?

Provide an appropriate response. -Consider the Gompertz function N=CaRt, where 0<R<1\mathrm { N } = \mathrm { Ca } \mathrm { R } ^ { \mathrm { t } } , \text { where } 0 < \mathrm { R } < 1N=CaRt, where 0<R<1 . What is the initial value of this function?

Functions, Graphs, and Models; Linear Functions

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Rewrite the expression as the sum and/or difference of logarithms, without using exponents. Simplify if possible. - $\log _ { n } \sqrt [ 7 ] { \frac { 6 x ^ { 5 } } { z ^ { 9 } } }$

Match the equation with its graph. - $y = 4 ^ { x + 5 }$

Provide an appropriate response. -Consider the logistic function $f ( x ) = \frac { c } { 1 + a e ^ { - b x } } , \text { where } b > 0$ 0. Is this function increasing or decreasing?

$2 ^ { ( 12 - 3 x ) } = 64$

Write the logarithmic equation in exponential form. - $\log _ { 10 } 0 = - 10$

Provide an appropriate response. -Consider the logistic function $f ( x ) = \frac { c } { 1 + a e ^ { - b x } } , \text { where } b > 0$ . What is the initial value of this function?

Find the exponential function that satisfies the given conditions. -Initial value $= 55$ , decreasing at a rate of $0.46 \%$ per week

The number of quarters needed to double an investment when a lump sum is invested at 4%, compounded quarterly, is given b $\mathrm { n } = \frac { \ln 2 } { \ln 1.01 }$ . In how many years will the investment double, to the nearest tenth of a year?

Explain the error in the following: $\log _ { 3 } 2 + \log _ { 3 } \mathrm { M } = \log _ { 3 } ( 2 + \mathrm { M } )$

Find the value of the logarithm without using a calculator. - $\log _ { 10 } 100$

$\mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { k } } \right) ^ { \mathrm { kn } } \text { for } \mathrm { P } = \$ 12,000 , \mathrm { n } = 6 , \mathrm { r } = 8 \% , \mathrm { k } = 4$

The number of periods needed to double an investment when a lump sum is invested at 2%, compounded semiannually, is given by n $\mathrm { n } = \log _ { 1.01 } 2$ 2. Use the change of base formula to find n, rounded to the nearest tenth.

Provide an appropriate response. -Using A $\mathrm { A } = \mathrm { P } ( 1 + r \mathrm { t } )$ ), the future value formula for a simple interest investment, derive the formula for t, the time (in years).

Use a change of base formula to evaluate the given logarithm. Approximate to three decimal places. - $\log _ { 8 } ( 0.662 )$

The number of periods needed to double an investment when a lump sum is invested at 11%, compounded semiannually, is given by n $\mathrm { n } = \frac { \ln 2 } { \ln 1.055 }$ . Find n, rounded to the nearest tenth.

$\mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { k } } \right) ^ { \mathrm { kn } } \text { for } \mathrm { P } = \$ 480 , \mathrm { n } = 3 , \mathrm { r } = 5 \% , \mathrm { k } = 4$

The number of quarters needed to double an investment when a lump sum is invested at 10%, compounded quarterly, is given b $\mathrm { n } = \log _ { 1.025 } 2$ In how many years will the investment double, to the nearest tenth of a Year?

Provide an appropriate response. -Consider the Gompertz function $\mathrm { N } = \mathrm { Ca } \mathrm { R } ^ { \mathrm { t } } , \text { where } 0 < \mathrm { R } < 1$ . What is the initial value of this function?