Free Printable Advanced Precalc MIDTERM 2024

1.

Find the inverse of the function f(x) = 2x - 1

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2.

Why are the graphs of 2^-xand (1/2)^x identical?

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3.

$Solve\ 3^{\left(x+2\right)}\ =\ \frac{1}{27}$

a)

x = 1

b)

x = -1

c)

x = -3

d)

x = -5

4.

Change x³ = 14 to a logarithm equation.

(write your answer as log base ... of ... equals ...)

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5.

Change ln x = 10 to an exponential equation.

a)

e^x = 10

b)

10^e = x

c)

log x = 10

d)

e^10 = x

6.

Find the exact value of $\log_381$

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7.

$Solve\ \log_4\ \left(3x-1\right)\ =\ \log_414$

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8.

$Solve\ \log_5\ \left(x\ +\ 20\right)\ +\ \log_5x\ =\ 3$

a)

x = 5

b)

x = -25 and x = 5

c)

x = 0

d)

x = 16

9.

What is the domain of f(x) = ln (x + 2)

a)

All reals.

b)

x > 0

c)

x > -2

d)

x < 2

10.

Use the change-of-base formula and calc to evaluate $\log_750$

a)

.497

b)

7.14

c)

2.01

d)

3.91

11.

Write (2 log x + 4 log 2) as a single log.

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12.

$\log_81\ \ +\ \log_88\ \ -\ \log_8\ \frac{1}{8}=$

a)

-1

b)

0

c)

1

d)

2

13.

What amount of money results from investing $1000 at 2.5% compounded monthly for 7 years?

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14.

What amount results from investing $500 at 4% compounded continuously after 8.5 years?

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15.

$If\ \log_8\ M\ =\ \frac{\log_{ }7}{\log_{ }8}\ \ then\ M\ =\$

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16.

Find the time needed to double an investment at a rate of 7% that is compounded continuously.

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17.

The size of a certain insect population at time t (in days) obeys the function P(t) = 750e^.03t . After how many days will the population reach 1500?

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18.

$P\left(t\right)=750e^{.03t}$ If an insect population follows that formula, after how many days will the population reach 2000?

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19.

Which is larger and why? Do not evaluate with a calculator. Explain using your knowledge of logs and exponents.
$\log_2\ 48\ \ or\ \log_2\ 64\ ?$

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20.

$\ \ \log_248\ \ or\ \log_348\$ Which is larger AND WHY?

No calculator needed!

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21.

3⁴ or e⁴ ? Which is larger and why?

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22.

Find the inverse of y = 1/(x-1).

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23.

Solve 10^x = 50

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24.

Solve 2 = log(x-3).

a)

x = 97

b)

x = 103

c)

x = 9.2

d)

x = .62

25.

Solve $\log_{\sqrt{2}}\ x\ =\ 4$

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26.

Solve 5^x = 4^x+1.

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27.

Solve for y: ln (y+4) = 5x + ln C. (C is a constant)

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28.

A piece of charcoal is found to contain 30% of the carbon-14 that it originally had. When did the tree from which the charcoal came die? (use 5600 years as the half-life of carbon-14.) (hint: find the decay rate first)

Growth/Decay Formula: A = A₀e^rt

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29.

How do the graphs of the exponential, y=2^x, and its inverse, y=log₂x, compare?

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30.

How does $f\left(x\right)\ =\ -3\cdot\log\ \left(x-1\right)$ compare to its' parent graph f(x) = log x?

a)

Three times as steep and shifted left 1 unit and upside down.

b)

1/3 as steep and upside down and shifted right 1 unit.

c)

Three times as steep and shifted right 1 unit and upside down.

d)

Three times as steep and moved down 1 unit.

31.

$\ \log_912\ \cdot\log_{12}15\cdot\log_{15}18$
This is an Extra Credit Problem: Write as a single log.

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32.

$f\left(x\right)\ =\frac{x^2\ +\ 3x\ -\ 4}{x^{2\ }+\ 2x\ -\ 3}$ List all the asymptotes, holes and x and y-intercepts for the rational function f(x).

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33.

$\frac{x^2}{\left(x\ -\ 7\right)\left(x+2\right)}$ What is the domain of this rational function?

a)

All reals

b)

$x\ \ne\ -7,\ 2$

c)

$x\ \ne7,\ -2$

d)

$x\ \ne7$

34.

$f\left(x\right)\ =\ \frac{x^3\ +\ 6x^2\ +\ 9}{x^{2\ }+\ 9}$

What is the domain of this function?

a)

x $\ne3,\ -3$

b)

x $\ne-3$

c)

All reals

d)

x $\ne0$

35.

$9^x\ -\ 3^x\ =$

a)

Cannot be simplified

b)

$6^x\ \ \ for\ all\ values\ of\ x$

c)

$3^x\ \ for\ all\ values\ of\ x$

d)

3

36.

$\frac{f\left(b\right)\ -\ f\left(a\right)}{b-a}$ is the formula for average rate of change of f(x) from [a, b].

What is the avg. rate of change of f(x) = ln x on the interval [4,6]?

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37.

Of the four types of savings account plans, which option would yield

exponential growth of the money in the account?

a)

Each successive year, 2% of the initial savings is added to the value of the account.

b)

Each successive year, 1.5% of the initial savings and $100 is added to the value of the account.

c)

Each successive year, 1% of the current value is added to the value of the account.

d)

Each successive year, $100 is added to the value of the account.

Advanced Precalc MIDTERM 2024

Total questions: 37

Worksheet time: 2hrs 12mins

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