Question 1

Solve for x.
-The radioactive decay of radon 222 can be described by the equation $\mathrm { A } = \mathrm { Ce } \mathrm { e } ^ { - 0.1813 \mathrm { t } }$, where $\mathrm { C }$ is the original amount of radon and $\mathrm { A }$ is the amount of radon after $\mathrm { t }$ days. If $2.5$ milligrams are in a laboratory container today, how much was there is the container 11 days ago? Round to the nearest hundredth.
A) $2.84 \mathrm { mg }$
B) $0.34 \mathrm { mg }$
C) $45.68 \mathrm { mg }$
D) $18.37 \mathrm { mg }$

Accepted Answer

The answer of Solve for x.
-The radioactive decay of radon...

Question 2

Solve.
-$$\log _ { 4 } \frac { 1 } { 64 } = x$$
A) $x = - 3$
B) $x = \frac { 1 } { 256 }$
C) $ x = \frac { 1 } { 16 }$
D) $x = 3$

Accepted Answer

The equation can be rewritten as $4^x = \frac{1}{64}$. Since $64 = 4^3$, we have $4^x = 4^{-3}$, which means $x = -3$.

Question 3

Write in exponential form.
-$$\log _ { 3 } 9 = x$$
A) $x = 2$
B) $x = 27$
C) $x = 12$
D) $x = 3$

Accepted Answer

The exponential form of $\log _ { 3 } 9 = x$ is $3^x = 9$. Since $3^2 = 9$, $x = 2$.

Question 4

Write in exponential form.
-$$\text { -3 = } \log _ { 10 } 0.001$$
A) $100.001 = - 3$
B) $0.001 ^ { - 3 } = 10$
C) $- 3^{10} = 0.001$
D) $10 ^ { - 3 } = 0.001$

Accepted Answer

The logarithmic equation $\log_{10} 0.001 = -3$ means that $10$ raised to the power of $-3$ equals $0.001$.

Question 5

Write in logarithmic form.
-$$343 = 7 ^ { 3 }$$
A) $\log _ { 7 } 343 = 3$
B) $\log _ { 3 } 343 = 7$
C) $\log _ { 7 } 3 = 343$
D) $\log _ { 343 } 7 = 3$

Accepted Answer

The logarithmic form of $343 = 7^3$ is $\log_7 343 = 3$, which means "the power to which the base 7 must be raised to produce 343 is 3."

Question 6

Write in logarithmic form.
-$$100,000 = 10 ^ { 5 }$$
A) $\log _ { 5 } 100,000 = 10$
B) $\log _ { 10 } 100,000 = 5$
C) $\log _ { 5 } 10 = 100,000$
D) $\log _ { 10 } 5 = 100,000$

Accepted Answer

The logarithmic form of $100,000 = 10^5$ is $\log_{10} 100,000 = 5$, meaning the power to which the base (10) must be raised to produce 100,000 is 5.

Question 7

Write in logarithmic form.
-$$8 = 2 ^ { 3 }$$
A) $\log _ { 8 } 2 = 3$
B) $\log _ { 2 } 3 = 8$
C) $\log _ { 3 } 8 = 2$
D) $\log _ { 2 } 8 = 3$

Accepted Answer

The logarithmic form of $8 = 2^3$ is $\log_{2}8 = 3$, which means "the exponent to which 2 must be raised to get 8 is 3".

Question 8

Write in logarithmic form.
-$$\frac { 1 } { 196 } = 14 ^ { - 2 }$$
A) $\log _ { 14 } \frac { 1 } { 196 } = - 2$
B) $\log _ { 14 } \frac { 1 } { 196 } = 14$
C) $\log _ { 14 } ( - 2 ) = \frac { 1 } { 196 }$
D) $\log \frac { 1 } { 196 } 14 = - 2$

Accepted Answer

The logarithmic form of $a^b = c$ is $\log_a c = b$. Therefore, the logarithmic form of $14^{-2} = \frac{1}{196}$ is $\log_{14} \frac{1}{196} = -2$.

Question 9

Write in exponential form.
-$$- 3 = \log _ { 2 } \frac { 1 } { 8 }$$
A) $2 ^ { 8 } = 3$
B) $2 ^ { - 3 } = \frac { 1 } { 8 }$
C) $\left( \frac { 1 } { 8 } \right) ^ { 3 } = 2$
D) $3 ^ { 2 } = \frac { 1 } { 8 }$

Accepted Answer

The exponential form of $\log _ { 2 } \frac { 1 } { 8 } = - 3$ is $2 ^ { - 3 } = \frac { 1 } { 8 }$, since $2$ raised to the power of $-3$ equals $\frac { 1 } { 8 }$.

Question 10

Write in logarithmic form.
-$$\frac { 1 } { 16 } = 4 ^ { - 2 }$$
A) $\log _ { 4 } \frac { 1 } { 16 } = - 2$
B) $\log _ { - 2 } \frac { 1 } { 16 } = 4$
C) $\log _ { 1 / 16 } 4 = - 2$
D) $\log _ { 4 } - 2 = \frac { 1 } { 16 }$

Accepted Answer

The logarithmic form of $\frac { 1 } { 16 } = 4 ^ { - 2 }$ is $\log _ { 4 } \frac { 1 } { 16 } = - 2$, which means "to what power must 4 be raised to get $\frac { 1 } { 16 }$?" The answer is -2.

Question 11

Write in exponential form.
-$$3 = \log _ { 5 } x$$
A) $5 ^ { x } = 3$
B) $5 ^ { 3 } = x$
C) $3 ^ { 5 } = x$
D) $x ^ { 3 } = 5$

Accepted Answer

The exponential form of $\log _ { 5 } x = 3$ is $5 ^ { 3 } = x$, which means the base (5) raised to the power on the right side of the equation (3) equals x.

Question 12

Solve for x.
-$$y = 180 ( 2.7 ) ^ { 0.07 t }$$
A) 402
B) 428
C) 415
D) 408

Accepted Answer

The given equation is an exponential function. To solve for x, we need to find the value of y when t is a specific value. The correct choice is based on the calculation of the expression for a given t, which results in approximately 415.

Question 13

Solve for x.
-The rabbit population in a forest area grows at the rate of 7% monthly. If there are 180 rabbits in September, find how many rabbits (rounded to the nearest whole number)should be expected by next September. Use The radioactive decay of radium 226 can be described by the equation $\mathrm { A } = \mathrm { Ce } - 0.0004279 \mathrm { t }$, where $\mathrm { C }$ is the original amount of radium and $\mathrm { A }$ is the amount of radium remaining after $\mathrm { t }$ years. If 22 milligrams of radium are sealed in a container now, how much radium will be in the container after 100 years. Round to the nearest thousand th.
A) $1.191 \mathrm { mg }$
B) $21.991 \mathrm { mg }$
C) $21.078 \mathrm { mg }$
D) $22.962 \mathrm { mg }$

Accepted Answer

The rabbit population grows at 7% monthly, so after 12 months, the population will be $180 \times (1.07)^{12} \approx 406$ rabbits. For the radium decay, using the formula $A = Ce^{-0.0004279 \times 100}$, with $C = 22$, we get $A \approx 21.078$ mg.

Question 14

Write in exponential form.
-$$- 6 = \log _ { 10 } ( 0.000001 )$$
A) $0.000001 ^ { - 6 } = 10$
B) $10 ^ { - 6 } = 0.000001$
C) $0.000001 ^ { 10 } = - 6$
D) $- 6^{10} = 0.000001$

Accepted Answer

The logarithmic equation $\log_{10}(0.000001) = -6$ means that $10$ raised to the power of $-6$ equals $0.000001$, which matches choice B.

Question 15

Write in exponential form.
-$$\frac { 1 } { 2 } = \log _ { 16 } 4$$
A) $16 ^ { 4 } = \frac { 1 } { 2 }$
B) $4 ^ { 1 / 2 } = 16$
C) $16 ^ { 1 / 2 } = 4$
D) $\left( \frac { 1 } { 2 } \right) ^ { 16 } = 4$

Accepted Answer

The exponential form of $\log _ { 16 } 4 = \frac { 1 } { 2 }$ is $16 ^ { 1 / 2 } = 4$, because the base (16) raised to the exponent ($\frac { 1 } { 2 }$) equals the number (4).

Question 16

Solve for x.
-A city is growing at the rate of $0.3 \%$ annually. If there were $5,226,000$ residents in the city in 2008 , find how many (to the nearest ten-thousand) are living in that city in 2012 . Use $y = 5,226,000 ( 2.7 ) ^ { 0.003 t }$.
A) $5,290,000$
B) $5,320,000$
C) $14,110,000$
D) 170,000

Accepted Answer

The correct formula for exponential growth is $y = P(1 + r)^t$, where $P$ is the initial amount, $r$ is the rate of growth, and $t$ is the time in years. However, the given formula uses a base of $2.7$ raised to $0.003t$, which seems to be a misinterpretation. Assuming a $0.3\%$ annual growth rate, the correct formula should be $y = 5,226,000(1 + 0.003)^t$. Since the growth rate in the question is mistakenly represented, but we proceed with the provided formula for calculation: $y = 5,226,000(2.7)^{0.003t}$. Given $t = 2012 - 2008 = 4$ years, we substitute $t$ in the formula. However, to correct the approach for an annual growth rate of $0.3\%$, we should use $y = 5,226,000(1.003)^4$. Calculating this correctly would not match the given formula, but aiming to match the closest answer based on the provided formula and the growth rate, we approximate the population after 4 years, which would be closest to $5,290,000$ when using a more standard approach to model a $0.3\%$ growth rate over 4 years.

Question 17

Write in exponential form.
-$$3 = \log _ { 5 } 125$$
A) $5 ^ { 3 } = 125$
B) $3 ^ { 5 } = 125$
C) $125 ^ { 3 } = 5$
D) $5 ^ { 125 } = 3$

Accepted Answer

The logarithmic equation $\log_5 125 = 3$ means that the base $5$ raised to the power $3$ equals $125$, which is correctly represented in exponential form as $5^3 = 125$.

Question 18

Write in logarithmic form.
-$$\mathrm { y } = \mathrm { e } ^ { - 2 }$$
A) $\log _ { e } - 2 = y$
B) $\log _ { y } e = - 2$
C) $\log _ { e } y = - 2$
D) $\log _ { - 2 } y = e$

Accepted Answer

The logarithmic form of $y = e^{-2}$ is $\log_e y = -2$, which is represented by choice C. This is because the base of the logarithm matches the base of the exponent (e), and the exponent becomes the value on the other side of the equation.

Question 19

Write in logarithmic form.
-$$0.001 = 10 ^ { - 3 }$$
A) $\log _ { - 3 } 10 = 0.001$
B) $\log _ { 10 } ( - 3 ) = 0.001$
C) $\log _ { 10 } ( 0.001 ) = - 3$
D) $\log _ { - 3 } ( 0.001 ) = 10$

Accepted Answer

The logarithmic form of $0.001 = 10 ^ { - 3 }$ is $\log _ { 10 } ( 0.001 ) = - 3$, where 10 is the base, 0.001 is the value, and -3 is the exponent.

Question 20

Write in exponential form.
-$$0 = \log _ { 13 } 1$$
A) $13 ^ { 0 } = 1$
B) $1 ^ { 13 } = 0$
C) $0 ^ { 13 } = 1$
D) does not exist

Accepted Answer

The logarithmic equation $\log_{13}1 = 0$ means that the base $13$ raised to the power $0$ equals $1$, which is correctly represented by $13^0 = 1$.

Quiz 12: Logarithmic and Exponential Functions

Solve. - $\log _ { 4 } \frac { 1 } { 64 } = x$

Write in exponential form. - $\log _ { 3 } 9 = x$

Write in exponential form. - $\text { -3 = } \log _ { 10 } 0.001$

Write in logarithmic form. - $343 = 7 ^ { 3 }$

Write in logarithmic form. - $100,000 = 10 ^ { 5 }$

Write in logarithmic form. - $8 = 2 ^ { 3 }$

Write in logarithmic form. - $\frac { 1 } { 196 } = 14 ^ { - 2 }$

Write in exponential form. - $- 3 = \log _ { 2 } \frac { 1 } { 8 }$

Write in logarithmic form. - $\frac { 1 } { 16 } = 4 ^ { - 2 }$

Write in exponential form. - $3 = \log _ { 5 } x$

Solve for x. - $y = 180 ( 2.7 ) ^ { 0.07 t }$

Write in exponential form. - $- 6 = \log _ { 10 } ( 0.000001 )$

Write in exponential form. - $\frac { 1 } { 2 } = \log _ { 16 } 4$

Solve for x. -A city is growing at the rate of $0.3 \%$ annually. If there were $5,226,000$ residents in the city in 2008 , find how many (to the nearest ten-thousand) are living in that city in 2012 . Use $y = 5,226,000 ( 2.7 ) ^ { 0.003 t }$ .

Write in exponential form. - $3 = \log _ { 5 } 125$

Write in logarithmic form. - $\mathrm { y } = \mathrm { e } ^ { - 2 }$

Write in logarithmic form. - $0.001 = 10 ^ { - 3 }$

Write in exponential form. - $0 = \log _ { 13 } 1$

Quiz 12: Logarithmic and Exponential Functions

Solve. - log⁡4164=x\log _ { 4 } \frac { 1 } { 64 } = xlog4​641​=x

Write in exponential form. - log⁡39=x\log _ { 3 } 9 = xlog3​9=x

Write in exponential form. - -3 = log⁡100.001\text { -3 = } \log _ { 10 } 0.001 -3 = log10​0.001

Write in logarithmic form. - 343=73343 = 7 ^ { 3 }343=73

Write in logarithmic form. - 100,000=105100,000 = 10 ^ { 5 }100,000=105

Write in logarithmic form. - 8=238 = 2 ^ { 3 }8=23

Write in logarithmic form. - 1196=14−2\frac { 1 } { 196 } = 14 ^ { - 2 }1961​=14−2

Write in exponential form. - −3=log⁡218- 3 = \log _ { 2 } \frac { 1 } { 8 }−3=log2​81​

Write in logarithmic form. - 116=4−2\frac { 1 } { 16 } = 4 ^ { - 2 }161​=4−2

Write in exponential form. - 3=log⁡5x3 = \log _ { 5 } x3=log5​x

Solve for x. - y=180(2.7) 0.07ty = 180 ( 2.7 ) ^ { 0.07 t }y=180(2.7) 0.07t

Write in exponential form. - −6=log⁡10(0.000001) - 6 = \log _ { 10 } ( 0.000001 ) −6=log10​(0.000001)

Write in exponential form. - 12=log⁡164\frac { 1 } { 2 } = \log _ { 16 } 421​=log16​4

Write in exponential form. - 3=log⁡51253 = \log _ { 5 } 1253=log5​125

Write in logarithmic form. - y=e−2\mathrm { y } = \mathrm { e } ^ { - 2 }y=e−2

Write in logarithmic form. - 0.001=10−30.001 = 10 ^ { - 3 }0.001=10−3

Write in exponential form. - 0=log⁡1310 = \log _ { 13 } 10=log13​1