Question 1

Solve.
-$\log _{5}\left(x^{2}-14 x+59\right)=\log _{5} 14$

Accepted Answer

A)  $\{-5,9\}$ 
B)  $\{5,9\}$. 
C)  $\{5\}$ 
D)  $\varnothing$ 
A)  $\{-5,9\}$ 
B)  $\{5,9\}$. 
C)  $\{5\}$ 
D)  $\varnothing$

Question 2

For the given function $f(x)$, find $f^{-1}(x)$. State the domain of $f^{-1}(x)$.
-$f(x)=(x-19)^{2}+4(x \geq 19)$

Accepted Answer

A)  $f^{1}(x)=\sqrt{x-4}+19, x \geq 4$ 
B)  $\mathrm{f}^{-1}(\mathrm{x})=4 \mathrm{x}^{2}+19, \mathrm{x} \geq 0$ 
C)  $\mathrm{f}^{-1}(\mathrm{x})=\sqrt{\mathrm{x}+4}-19, \mathrm{x} \geq-4$ 
D)  $f^{-1}(x)=\sqrt{x+19}-4, x \geq-19$ 
A)  $f^{1}(x)=\sqrt{x-4}+19, x \geq 4$ 
B)  $\mathrm{f}^{-1}(\mathrm{x})=4 \mathrm{x}^{2}+19, \mathrm{x} \geq 0$ 
C)  $\mathrm{f}^{-1}(\mathrm{x})=\sqrt{\mathrm{x}+4}-19, \mathrm{x} \geq-4$ 
D)  $f^{-1}(x)=\sqrt{x+19}-4, x \geq-19$

Question 3

Rewrite in exponential form.
-$\ln x=2$

Accepted Answer

A)  $e^{x}=2$ 
B)  $10^{\mathrm{x}}=2$ 
C)  $10^{2}=\mathrm{x}$ 
D)  $e^{2}=x$ 
A)  $e^{x}=2$ 
B)  $10^{\mathrm{x}}=2$ 
C)  $10^{2}=\mathrm{x}$ 
D)  $e^{2}=x$

Question 4

Use the formula $ \frac{f(b)-f(a)}{b-a} $ to find the average rate of change of the function f(x) on the interval [a, b].
-$f(x)=4^{x},[2,5]$

Accepted Answer

A)  336 
B)  504 
C)  208 
D)  347 
A)  336 
B)  504 
C)  208 
D)  347

Question 5

The sales of a mature product (one which has passed its peak) will decline according to the function $\mathrm{S}(\mathrm{t})=\mathrm{S}_{\mathrm{O}} \mathrm{e}^{- \text {at }}$, where $\mathrm{t}$ is time in years since the peak sales. Find the sales of a product 22 years after its peak sales if $\mathrm{a}=0.13$ and $\mathrm{So}=15,100$.

Accepted Answer

A)  17,196 
B)  865 
C)  300,274 
D)  433 
A)  17,196 
B)  865 
C)  300,274 
D)  433

Question 6

Compute the compound interest.
-How long will it take for $\$ 3300$ to grow to $\$ 14,800$ at an interest rate of $4.1 \%$ if the interest is compounded quarterly? Round the number of years to the nearest hundredth.

Accepted Answer

A)  $109.95 \mathrm{yr}$ 
B)  $37.35 \mathrm{yr}$ 
C)  $36.79 \mathrm{yr}$ 
D)  $147.16 \mathrm{yr}$ 
A)  $109.95 \mathrm{yr}$ 
B)  $37.35 \mathrm{yr}$ 
C)  $36.79 \mathrm{yr}$ 
D)  $147.16 \mathrm{yr}$

Question 7

Solve.
-$\left(\frac{1}{5}\right)^{6 x-14}=\frac{1}{625}$

Accepted Answer

A)  $\{-3\}$ 
B)  $\{3\}$ 
C)  $\{4\}$ 
D)  $\{-4\}$ 
A)  $\{-3\}$ 
B)  $\{3\}$ 
C)  $\{4\}$ 
D)  $\{-4\}$

Question 8

Evaluate the given function
-$f(x)=3^{x}, f(-4)$

Accepted Answer

A)  12 
B)  $\frac{1}{81}$ 
C)  $-\frac{3}{4}$ 
D)  $\frac{1}{12}$ 
A)  12 
B)  $\frac{1}{81}$ 
C)  $-\frac{3}{4}$ 
D)  $\frac{1}{12}$

Question 9

For the given logarithmic function, find a) the average rate of change for the interval $[2,6]$ and b) the interval $[3,5]$, as well as $c$ ) the rate of change at $x=4$. Round all calculations to the nearest thousandth.
-$f(x)=\ln (x+1)$

Accepted Answer

A)  a) 0.275 b) 0.255 c) 0.2 
B)  a) 0.212 b) 0.203 c) 0.2 
C)  a) 0.2 b) 0.255 c) 0.275 
D)  a) 0.2 b) 0.203 c) 0.212 
A)  a) 0.275 b) 0.255 c) 0.2 
B)  a) 0.212 b) 0.203 c) 0.2 
C)  a) 0.2 b) 0.255 c) 0.275 
D)  a) 0.2 b) 0.203 c) 0.212

Question 10

Rewrite as a single logarithm using the quotient rule for logarithms. Assume all variables represent positive real numbers.
-$\log _{6} 9-\log _{6} y$

Accepted Answer

A)  $\log _{6}(9-y)$ 
B)  $\log _{12} \frac{9}{y}$ 
C)  $\log _{6} \frac{9}{y}$ 
D)  $\log _{6} \frac{y}{9}$ 
A)  $\log _{6}(9-y)$ 
B)  $\log _{12} \frac{9}{y}$ 
C)  $\log _{6} \frac{9}{y}$ 
D)  $\log _{6} \frac{y}{9}$

Question 11

Determine whether the graph is the graph of a function.
-

Accepted Answer

A) True 
 B)False

Question 12

Evaluate the given function.
-$f(x)=\log _{3} x, f(1)$

Accepted Answer

A)  3 
B)  -3 
C)  0 
D)  1 
A)  3 
B)  -3 
C)  0 
D)  1

Question 13

Evaluate the given function
-$f(x)=\left(\frac{1}{5}\right)^{x}, f(-2)$

Accepted Answer

A)  $\frac{1}{25}$ 
B)  $-\frac{1}{25}$ 
C)  -10 
D)  25 
A)  $\frac{1}{25}$ 
B)  $-\frac{1}{25}$ 
C)  -10 
D)  25

Question 14

Given $g(x)$ and $(g \circ f)(x)$, find $f(x)$.
-$g(x)=\frac{3}{4} x+2,(g \circ f)(x)=12 x+23$

Accepted Answer

A)  $f(x)=12 x+20$ 
B)  $f(x)=8 x+14$ 
C)  $f(x)=18 x+32$ 
D)  $f(x)=16 x+28$ 
A)  $f(x)=12 x+20$ 
B)  $f(x)=8 x+14$ 
C)  $f(x)=18 x+32$ 
D)  $f(x)=16 x+28$

Question 15

Evaluate. Round to the nearest thousandth, if necessary.
-$\ln 284$

Accepted Answer

A)  0.176 
B)  104.797 
C)  2.453 
D)  5.649 
A)  0.176 
B)  104.797 
C)  2.453 
D)  5.649

Question 16

Rewrite using the power rule. Assume all variables represent positive real numbers.
-$\log _{3} 3^{-20}$

Accepted Answer

A)  -60 
B)  3 
C)  -20 
D)  1 
A)  -60 
B)  3 
C)  -20 
D)  1

Question 17

Evaluate the given function. Round to the nearest thousandth.
-$f(x)=e^{2 x+3}, f(1.9)$

Accepted Answer

A)  897.847 
B)  2.226 
C)  2.460 
D)  $120,571.715$ 
A)  897.847 
B)  2.226 
C)  2.460 
D)  $120,571.715$

Question 18

Solve. Round to the nearest thousandth, if necessary
-$10^{6 x+15}=48,994$

Accepted Answer

A)  $\{118.141\}$ 
B)  $\{-61.859\}$ 
C)  $\{3.282\}$ 
D)  $\{-1.718\}$ 
A)  $\{118.141\}$ 
B)  $\{-61.859\}$ 
C)  $\{3.282\}$ 
D)  $\{-1.718\}$

Question 19

Determine whether the functions $f(x)$ and $g(x)$ are inverse functions.
-$f(x)=-7 x, g(x)=\frac{1}{7} x$

Accepted Answer

A) True 
 B)False

Question 20

Solve the problem.
-The amount of money, in billions of dollars, spent on health care that was covered by insurance in a certain country in a particular year can be approximated by the function $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}+19 \mathrm{x}+293$, where $x$ represents the number of years after 1995. The amount of money, in billions of dollars, spent on health care that was paid out of pocket in a certain country in a particular year can be approximated by the function $\mathrm{g}(\mathrm{x})=5 \mathrm{x}+142$, where again $\mathrm{x}$ represents the number of years after 1995.
(i) Find ( $\mathrm{f}-\mathrm{g})(\mathrm{x})$. Explain, in your own words, what this function represents.
(ii) Find ( $\mathrm{f}-\mathrm{g})(55)$. Explain, in your own words, what this number represents.

Accepted Answer

A)  (i) ( $\mathrm{f}-\mathrm{g})(\mathrm{x})=\mathrm{x}^{2}+14 \mathrm{x}+151$; this is the difference between health- care costs covered by insurance and health- care costs that are paid out of pocket, in billions of dollars, $x$ years after 1995.
(ii) 3946; health- care costs covered by insurance will be $\$ 3,946,000,000,000$ higher than health- care costs that are paid out of pocket in 2050. 
B)  (i) (f - g) $(x)=x^{2}-24 x-435$; this is the difference between health- care costs covered by insurance and health- care costs that are paid out of pocket, in billions of dollars, $\mathrm{x}$ years after 1995.
(ii) 1270; health- care costs covered by insurance will be $\$ 1,270,000,000,000$ higher than health- care costs that are paid out of pocket in 2050. 
C)  (i) (f - g)(x) $=x^{2}-24 x-435$; this is the difference between health- care costs that are paid out of pocket and those that are covered by insurance, in billions of dollars, $\mathrm{x}$ years after 1995.
(ii) 1270; health- care costs covered by insurance will be $\$ 1,270,000,000,000$ less than heal th- care costs that are paid out of pocket in 2050. 
D)  (i) (f - g) $(x)=x^{2}+14 x+151$; this is the difference between health- care costs that are paid out of pocket and those that are covered by insurance, in billions of dollars, $\mathrm{x}$ years after 1995.
(ii) 3946; health- care costs covered by insurance will be $\$ 3,946,000,000,000$ less than heal th- care costs that are paid out of pocket in 2050. 
A)  (i) ( $\mathrm{f}-\mathrm{g})(\mathrm{x})=\mathrm{x}^{2}+14 \mathrm{x}+151$; this is the difference between health- care costs covered by insurance and health- care costs that are paid out of pocket, in billions of dollars, $x$ years after 1995.
(ii) 3946; health- care costs covered by insurance will be $\$ 3,946,000,000,000$ higher than health- care costs that are paid out of pocket in 2050. 
B)  (i) (f - g) $(x)=x^{2}-24 x-435$; this is the difference between health- care costs covered by insurance and health- care costs that are paid out of pocket, in billions of dollars, $\mathrm{x}$ years after 1995.
(ii) 1270; health- care costs covered by insurance will be $\$ 1,270,000,000,000$ higher than health- care costs that are paid out of pocket in 2050. 
C)  (i) (f - g)(x) $=x^{2}-24 x-435$; this is the difference between health- care costs that are paid out of pocket and those that are covered by insurance, in billions of dollars, $\mathrm{x}$ years after 1995.
(ii) 1270; health- care costs covered by insurance will be $\$ 1,270,000,000,000$ less than heal th- care costs that are paid out of pocket in 2050. 
D)  (i) (f - g) $(x)=x^{2}+14 x+151$; this is the difference between health- care costs that are paid out of pocket and those that are covered by insurance, in billions of dollars, $\mathrm{x}$ years after 1995.
(ii) 3946; health- care costs covered by insurance will be $\$ 3,946,000,000,000$ less than heal th- care costs that are paid out of pocket in 2050.

Solve. - $\log _{5}\left(x^{2}-14 x+59\right)=\log _{5} 14$

For the given function $f(x)$ , find $f^{-1}(x)$ . State the domain of $f^{-1}(x)$ . - $f(x)=(x-19)^{2}+4(x \geq 19)$

Rewrite in exponential form. - $\ln x=2$

Use the formula $\frac{f(b)-f(a)}{b-a}$ to find the average rate of change of the function f(x) on the interval [a, b]. - $f(x)=4^{x},[2,5]$

Compute the compound interest. -How long will it take for $\$ 3300$ to grow to $\$ 14,800$ at an interest rate of $4.1 \%$ if the interest is compounded quarterly? Round the number of years to the nearest hundredth.

Solve. - $\left(\frac{1}{5}\right)^{6 x-14}=\frac{1}{625}$

Evaluate the given function - $f(x)=3^{x}, f(-4)$

For the given logarithmic function, find a) the average rate of change for the interval $[2,6]$ and b) the interval $[3,5]$ , as well as $c$ ) the rate of change at $x=4$ . Round all calculations to the nearest thousandth. - $f(x)=\ln (x+1)$

Rewrite as a single logarithm using the quotient rule for logarithms. Assume all variables represent positive real numbers. - $\log _{6} 9-\log _{6} y$

Determine whether the graph is the graph of a function. -

Evaluate the given function. - $f(x)=\log _{3} x, f(1)$

Evaluate the given function - $f(x)=\left(\frac{1}{5}\right)^{x}, f(-2)$

Given $g(x)$ and $(g \circ f)(x)$ , find $f(x)$ . - $g(x)=\frac{3}{4} x+2,(g \circ f)(x)=12 x+23$

Evaluate. Round to the nearest thousandth, if necessary. - $\ln 284$

Rewrite using the power rule. Assume all variables represent positive real numbers. - $\log _{3} 3^{-20}$

Evaluate the given function. Round to the nearest thousandth. - $f(x)=e^{2 x+3}, f(1.9)$

Solve. Round to the nearest thousandth, if necessary - $10^{6 x+15}=48,994$

Determine whether the functions $f(x)$ and $g(x)$ are inverse functions. - $f(x)=-7 x, g(x)=\frac{1}{7} x$

Review of Real Numbers

Linear and Absolute Value Equations and Inequalities

Graphing Linear Equations

Systems of Equations

Exponents, Polynomials, and Factoring Polynomials

Rational Expressions and Equations

Radical Expressions and Equations

Quadratic Equations

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Exam 9: Logarithmic and Exponential Functions

Solve. - log⁡5(x2−14x+59)=log⁡514\log _{5}\left(x^{2}-14 x+59\right)=\log _{5} 14log5​(x2−14x+59)=log5​14

For the given function f(x)f(x)f(x) , find f−1(x)f^{-1}(x)f−1(x) . State the domain of f−1(x)f^{-1}(x)f−1(x) . - f(x)=(x−19)2+4(x≥19)f(x)=(x-19)^{2}+4(x \geq 19)f(x)=(x−19)2+4(x≥19)

Rewrite in exponential form. - ln⁡x=2\ln x=2lnx=2

Use the formula f(b)−f(a)b−a \frac{f(b)-f(a)}{b-a} b−af(b)−f(a)​ to find the average rate of change of the function f(x) on the interval [a, b]. - f(x)=4x,[2,5]f(x)=4^{x},[2,5]f(x)=4x,[2,5]

Compute the compound interest. -How long will it take for $3300\$ 3300$3300 to grow to $14,800\$ 14,800$14,800 at an interest rate of 4.1%4.1 \%4.1% if the interest is compounded quarterly? Round the number of years to the nearest hundredth.

Solve. - (15)6x−14=1625\left(\frac{1}{5}\right)^{6 x-14}=\frac{1}{625}(51​)6x−14=6251​

Evaluate the given function - f(x)=3x,f(−4)f(x)=3^{x}, f(-4)f(x)=3x,f(−4)

For the given logarithmic function, find a) the average rate of change for the interval [2,6][2,6][2,6] and b) the interval [3,5][3,5][3,5] , as well as ccc ) the rate of change at x=4x=4x=4 . Round all calculations to the nearest thousandth. - f(x)=ln⁡(x+1)f(x)=\ln (x+1)f(x)=ln(x+1)

Rewrite as a single logarithm using the quotient rule for logarithms. Assume all variables represent positive real numbers. - log⁡69−log⁡6y\log _{6} 9-\log _{6} ylog6​9−log6​y