Mid-Term Exam 3

Evaluate the expression. (Simplify your answer completely.) (a) logbase7(49) (b) logbase8(64) (c) logbase9(9^10)

(a) 2 (b) 2 (c) 10

Evaluate the expression. (Simplify your answer completely.) (a) e^ln(√3) (b) e^ln(1/𝜋) (c) 10^log(16)

(a) square root of 3 (b) 1/pi (c) 16

-2^x

Growth below x-axis

If given the point (2, 1/4), what is the function?

(1/2)^x

A man invests $3000 in an account that pays 6.5% interest per year, compounded quarterly. (a) Find the amount after 3 years? (Round your answer to the nearest cent.) (b)How long will it take for the investment to triple? (Round your answer to two decimal places.)

(a) $3640.22 (b) 17.04 years

If $3000 is invested at an interest rate of 9.25% per year, compounded continuously, find the value of the investment after the given number of years. (Round your answers to the nearest cent.) (a) 4 years (b) 8 years (c) 12 years

(a) $4343.20 (b) $6287.81 (c) $9103.08

If $5000 is borrowed at a rate of 6.75% interest per year, compounded quarterly, find the amount due at the end of the given number of years. (Round your answers to the nearest cent.) (a) 3 years (b) 5 years (c) 7 years

(a) $6111.96 (b) $6987.49 (c) $7988.44

20/1-e^-x = 2 (a) Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places.

(a) -ln(9) (b) x = -2.197225

3(1 + 10^6x) = 8 (a) Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places.

(a) 1/6log(5/3) (b) x = 0.036975

Use the definition of the logarithmic function to find x. (Simplify your answer completely.) (a) logbase3(x) = −2 (b) logbase5(125) = x

(a) 1/9 (b) 3

A bacteria culture contains 1500 bacteria initially and doubles every hour. (a) Find a function N that models the number of bacteria after t hours. (b) Find the number of bacteria after 24 hours.

(a) 1500(2)^t (b) 25,165,824,000 bacteria

Express the equation in exponential form. (a) ln(3) = 2y (b) ln(t + 1) = −1

(a) 3=e^{2y} (b) t+1=e^{-1}

The population of the world was 7.1 billion in 2013, and the observed relative growth rate was 1.1% per year. (a) Estimate how long it takes the population to double. (Round your answer to two decimal places.) (b) Estimate how long it takes the population to triple. (Round your answer to two decimal places.)

(a) 63.01 years (b) 99.87 years

e^(7 − 8x) = 18 (a) Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places.

(a) x = ln(18) - 7/ -8 (b) x = 0.513704

3^(4x − 1) = 8 (a) Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places.

(a) x = logbase3(8) + 1/4 (b) x = 0.723197

Express the equation in logarithmic form. (a) 4−1/2 = 0.5 (b) 83 = 512

(a)logbase4(0.5)=-1/2 (b)logbase8(512)=3

(1/3)^x Make a table of values and graph the function.

-2 = 9 -1 = 3 0 = 1 1 = 1/3 2 = 1/9

An investment of $3000 is deposited into an account in which interest is compounded monthly. Complete the table by filling in the amounts to which the investment grows at the indicated interest rates. (Round your answers to the nearest cent.) t = 5 yr

1% = $3153.75 2% = $3315.24 3% = $3484.85 4% = $3662.99 5% = $3850.08 6% = $4046.55

Use the Laws of Logarithms to expand the expression. log(square root of t^7)

7/2log(t)

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. logbase2(577)

9.172428

How long will it take for an investment of $1000 to double in value if the interest rate is 7.5% per year, compounded continuously? (Round your answer to two decimal places.)

9.24 years

2^-x

Decay above x-axis

-2^-x

Decay below the x-axis

Graph the. following function and state the domain, range, and asymptote. 6-5^x

Domain = (-infinity, +infinity) Range = (-infinity, 6) Asymptote = y=6

Graph the following function and state the domain, range, and asymptote. 9-e^x

Domain = (-infinity, +infinity) Range = (-infinity, 9) Asymptote = y=9

Graph the following function and state the domain, range, and asymptote. y=e^(x-2)+4

Domain = (-infinity, +infinity) Range = (4, +infinity) Asymptote = y=4

Find the domain of the function. (Enter your answer using interval notation.) g(x) = logbase4(x^2 − 9)

Domain = (-infinity, -3) U (3, +infinity)

Graph the following function and state the domain, range, and asymptote. y=logbase3(-x)

Domain = (-infinity, 0) Range = (-infinity, +infinity) Asymptote = x=0

Find the domain of the function. (Enter your answer using interval notation.) f(x) = logbase9(6 − 3x)

Domain = (-infinity, 2)

2^x

growth above x-axis

-logbase2(-x)

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-logbase2(x)

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logbase2(-x)

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logbase2(x)

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Use the Laws of Logarithms to combine the expression. 5 ln(2) + 2 ln(x) − 1/2 ln(x + 5)

ln(32/square root of (x+5))

Use the Laws of Logarithms to expand the expression. ln(r/3s)

ln(r)-[ln(3)+ln(s)]

Use the Laws of Logarithms to combine the expression. 3(logbase7(x) + 2 logbase7(y) − 4 logbase7(z))

logbase7(x^3y^6/z^12)

Solve the equation. (Enter your answers as a comma-separated list. Round your answers to four decimal places.) 4^16x − 4^8x − 12 = 0

x = 1/8

Solve the logarithmic equation for x, as in Example 7. (Enter your answers as a comma-separated list.) ln(6 + x) = 3

x = 14.0855

Solve the exponential equation 2e^x = 40.

x = 2.996

Solve the logarithmic equation for x, as in Example 7. (Enter your answers as a comma-separated list.) log(x) + log(x − 1) = log(2x)

x = 3

Solve the logarithmic equation for x. (Enter your answers as a comma-separated list.) log(x) + log(x − 3) = 1

x = 5

Solve the logarithmic equation for x, as in Example 7. (Enter your answers as a comma-separated list.) 2 log(x) = log(2) + log(4x − 6)

x = 6, 2

Solve the logarithmic equation for x. (Enter your answers as a comma-separated list.) logbase2(x^2 − 3x − 32) = 3

x = 8, -5