Question 1

Write the equation in logarithmic form. $$5 ^ { - 5 } = \frac { 1 } { 3125 }$$
A) $\log _ { 5 } \frac { 1 } { 3125 } = - 5$
B) $\log _ { - 5 } 5 = 3125$
C) $\log _ { 5 } ( - 5 ) = 3125$
D) $- \log _ { 5 } 3125 = - 5$

Accepted Answer

The logarithmic form of $a^b = c$ is $\log_a c = b$. Given $5^{-5} = \frac{1}{3125}$, the equivalent logarithmic form is $\log_5 \frac{1}{3125} = -5$.

Question 2

Write the equation in exponential form. $$\log _ { 4 } 4096 = 6$$
A) $4 ^ { 6 } = 4096$
B) $6 ^ { 4096 } = 4$
C) $6 ^ { 4 } = 4096$
D) $4096 ^ { 6 } = 4$

Accepted Answer

The exponential form of $\log _ { 4 } 4096 = 6$ is $4 ^ { 6 } = 4096$, which matches choice A. This is because the base of the logarithm (4) raised to the power on the right side of the equation (6) equals the number inside the logarithm (4096).

Question 3

Evaluate without the use of a calculator. $\log 0.001$
A) 2
B) $0.001$
C) $- 3$
D) $\frac { 1 } { 3 }$

Accepted Answer

$\log 0.001 = \log 10^{-3} = -3$ because the base 10 logarithm of a number is the exponent to which 10 must be raised to produce that number.

Question 4

Evaluate without the use of a calculator. $\log 10 ^ { 5 }$
A) 5
B) 50
C) $5 ^ { 10 }$
D) $- 5$

Accepted Answer

The logarithm of a number is the exponent to which the base must be raised to produce that number. Here, the base is 10, and we want to know to what power 10 must be raised to get $10^5$. The answer is 5, because $10^5 = 10^5$.

Question 5

Which is a logarithmic function? 
A) $h ( x ) = \sqrt { 20 x + 12 }$
B) $g ( x ) = \log _ { 20 } x$
C) $f ( x ) = 20 ^ { x + 2 }$
D) $k ( x ) = \frac { 5 x } { x - 20 }$

Accepted Answer

A logarithmic function is defined as a function that includes a logarithm of a variable. Choice B, $g ( x ) = \log _ { 20 } x$, is the only option that fits this definition, as it involves the logarithm of $x$ with base 20.

Question 6

In 2000, the population of Sheboygan, WI was about 113,000 and was growing at a rate of 0.85% per year. Use an exponential function to predict the population in 2021 to the nearest whole Number.
A) 134,981
B) 128,481
C) 460,982,820
D) 145,034

Accepted Answer

The population can be predicted using the formula $P = P_0(1 + r)^t$, where $P_0$ is the initial population, $r$ is the growth rate, and $t$ is the time in years. Plugging in the values: $P = 113,000(1 + 0.0085)^{21}$, which calculates to approximately 134,981.

Question 7

Evaluate without the use of a calculator. $$\log _ { 8 } \sqrt [ 5 ] { 8 }$$
A) $\frac { 1 } { 5 }$
B) 1
C) 5
D) $- 5$

Accepted Answer

The expression $\log _ { 8 } \sqrt [ 5 ] { 8 }$ simplifies to $\log _ { 8 } 8^{1/5}$. Since $\log _ { b } b^{x} = x$, the answer is $1/5$.

Question 8

The population of bacteria culture was 2000 at noon, and was increasing at a rate of 10% per hour. The number can be found using the function $P ( t ) = 2,000 ( 1.1 ) ^ { t }$ where t is the number of hours past noon. Predict the population 9 hours later, at 9 PM to the nearest Whole number.
A) 4716
B) 19,800
C) 2716
D) 6736

Accepted Answer

The population 9 hours later can be found by substituting $t = 9$ into the given function: $P(9) = 2000(1.1)^9$. Calculating this gives approximately 4716.

Question 9

Evaluate without the use of a calculator. $\log \left( 1 \times 10 ^ { 5 } \right)$
A) 5
B) 6
C) $- 5$
D) $\frac { 1 } { 5 }$

Accepted Answer

The logarithm of a number in base 10 is the exponent to which 10 must be raised to produce that number. Here, $10^5 = 100,000$, so $\log(10^5) = 5$.

Question 10

Evaluate without the use of a calculator. $$\log _ { 2 / 3 } \frac { 8 } { 27 }$$
A) 3
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 4 } { 9 }$

Accepted Answer

The expression can be rewritten as $\log_{\frac{2}{3}}\left(\frac{2^3}{3^3}\right)$ or $\log_{\frac{2}{3}}\left(\left(\frac{2}{3}\right)^{-3}\right)$. The base and the argument are inverses of each other raised to the power of -3, thus the answer is 3.

Question 11

In 2000, the population of Sheboygan, WI was about 113,000 and was growing at a rate of 0.85% per year. Find an exponential function that describes the population P in terms of t, the number of Years after 2000. 
A) $P ( t ) = 113,000 + 1.85 ^ { t }$
B) $P ( t ) = 113,000 ( 1.0085 ) ^ { t }$
C) $P ( t ) = 113,000 ( 1.85 ) ^ { t }$
D) $P ( t ) = 0.85 ( 113,000 )$

Accepted Answer

The correct formula for exponential growth is $P(t) = P_0(1 + r)^t$, where $P_0$ is the initial population, $r$ is the growth rate as a decimal, and $t$ is the time in years. Therefore, the correct formula is $P(t) = 113,000(1.0085)^t$, which matches choice B.

Question 12

Scientists often use a process called carbon dating to estimate the age of archaeological finds. The process measures the amount of carbon-14, a radioactive isotope with a half-life of 5730 years. If a sample of wood from an ancient artifact had 13 grams of carbon-14 initially, the amount remaining, in grams, is given by $A ( t ) = 13 \left( \frac { 1 } { 2 } \right) ^ { t / 5730 }$ where t is the number of years since the tree died. How many grams would be present after 22,920 years?

Accepted Answer

The answer of Scientists often use a process called carbon...

Question 13

Write the equation in exponential form. $$y = \log _ { b } x$$
A) $b ^ { x } = y$
B) $y ^ { x } = b$
C) $y ^ { b } = x$
D) $b ^ { y } = x$

Accepted Answer

The exponential form of the logarithmic equation $y = \log_b x$ is $b^y = x$. This is because the base $b$ raised to the power of the logarithm's result $y$ equals the logarithm's argument $x$.

Question 14

Evaluate without the use of a calculator. $\log _ { 4 } 4$

Accepted Answer

The answer of Evaluate without the use of a calculator....

Question 15

Evaluate without the use of a calculator. $\log _ { 6 } \frac { 1 } { 36 }$
A) -2
B) 4
C) 12
D) 2

Accepted Answer

The logarithm $\log _ { 6 } \frac { 1 } { 36 }$ can be rewritten as $\log _ { 6 } 6^{-2}$, which simplifies to -2 because the base of the logarithm and the base of the exponent are the same.

Question 16

Write the equation in exponential form. $\log _ { 17 / 5 } 1 = 0$
A) $\left( \frac { 17 } { 5 } \right) ^ { 0 } = 1$
B) $\frac { 17 ^ { 1 } } { 5 } = 0$
C) $1 ^ { 0 } = \frac { 17 } { 5 }$
D) $\frac { 17 ^ { 0 } } { 5 } = 1$

Accepted Answer

The exponential form of $\log _ { b } a = c$ is $b^c = a$. Here, $b = \frac{17}{5}$, $a = 1$, and $c = 0$, so the correct transformation is $\left( \frac { 17 } { 5 } \right) ^ { 0 } = 1$.

Question 17

Evaluate without the use of a calculator. $$\log _ { t } \frac { 1 } { t }$$

Accepted Answer

The answer of Evaluate without the use of a calculator....

Question 18

Evaluate without the use of a calculator. $\log _ { t } t ^ { 19 }$
A) $\frac { 1 } { 19 }$
B) 19
C) $- 19$
D) 1

Accepted Answer

The logarithm of a number to its own base is always the exponent to which the base must be raised to produce that number. Therefore, $\log_t(t^{19}) = 19$.

Question 19

Write the equation in logarithmic form. $$4 ^ { 3 } = 64$$
A) $\log _ { 4 } 64 = 3$
B) $\log _ { 64 } 4 = 3$
C) $\log _ { 3 } 4 = 64$
D) $\log _ { 4 } 3 = 64$

Accepted Answer

The logarithmic form of $4^3 = 64$ is $\log_{4}64 = 3$, which means "the exponent to which 4 must be raised to get 64 is 3".

Question 20

Write the equation in exponential form. $$\log _ { 3 } \frac { 1 } { 729 } = - 6$$

Accepted Answer

The answer of Write the equation in exponential form. \[\log...

Quiz 8: Exponential and Logarithmic Functions and Applications

Write the equation in logarithmic form. $5 ^ { - 5 } = \frac { 1 } { 3125 }$

Write the equation in exponential form. $\log _ { 4 } 4096 = 6$

Evaluate without the use of a calculator. $\log 0.001$

Evaluate without the use of a calculator. $\log 10 ^ { 5 }$

Which is a logarithmic function?

In 2000, the population of Sheboygan, WI was about 113,000 and was growing at a rate of 0.85% per year. Use an exponential function to predict the population in 2021 to the nearest whole Number.

Evaluate without the use of a calculator. $\log _ { 8 } \sqrt [ 5 ] { 8 }$

The population of bacteria culture was 2000 at noon, and was increasing at a rate of 10% per hour. The number can be found using the function $P ( t ) = 2,000 ( 1.1 ) ^ { t }$ where t is the number of hours past noon. Predict the population 9 hours later, at 9 PM to the nearest Whole number.

Evaluate without the use of a calculator. $\log \left( 1 \times 10 ^ { 5 } \right)$

Evaluate without the use of a calculator. $\log _ { 2 / 3 } \frac { 8 } { 27 }$

In 2000, the population of Sheboygan, WI was about 113,000 and was growing at a rate of 0.85% per year. Find an exponential function that describes the population P in terms of t, the number of Years after 2000.

Write the equation in exponential form. $y = \log _ { b } x$

Evaluate without the use of a calculator. $\log _ { 4 } 4$

Evaluate without the use of a calculator. $\log _ { 6 } \frac { 1 } { 36 }$

Write the equation in exponential form. $\log _ { 17 / 5 } 1 = 0$

Evaluate without the use of a calculator. $\log _ { t } \frac { 1 } { t }$

Evaluate without the use of a calculator. $\log _ { t } t ^ { 19 }$

Write the equation in logarithmic form. $4 ^ { 3 } = 64$

Write the equation in exponential form. $\log _ { 3 } \frac { 1 } { 729 } = - 6$

Quiz 8: Exponential and Logarithmic Functions and Applications

Write the equation in logarithmic form. 5−5=131255 ^ { - 5 } = \frac { 1 } { 3125 }5−5=31251​

Write the equation in exponential form. log⁡44096=6\log _ { 4 } 4096 = 6log4​4096=6

Evaluate without the use of a calculator. log⁡0.001\log 0.001log0.001

Evaluate without the use of a calculator. log⁡105\log 10 ^ { 5 }log105

Which is a logarithmic function?

In 2000, the population of Sheboygan, WI was about 113,000 and was growing at a rate of 0.85% per year. Use an exponential function to predict the population in 2021 to the nearest whole Number.

Evaluate without the use of a calculator. log⁡885\log _ { 8 } \sqrt [ 5 ] { 8 }log8​58​

Evaluate without the use of a calculator. log⁡(1×105) \log \left( 1 \times 10 ^ { 5 } \right) log(1×105)

Evaluate without the use of a calculator. log⁡2/3827\log _ { 2 / 3 } \frac { 8 } { 27 }log2/3​278​

In 2000, the population of Sheboygan, WI was about 113,000 and was growing at a rate of 0.85% per year. Find an exponential function that describes the population P in terms of t, the number of Years after 2000.

Write the equation in exponential form. y=log⁡bxy = \log _ { b } xy=logb​x

Evaluate without the use of a calculator. log⁡44\log _ { 4 } 4log4​4

Evaluate without the use of a calculator. log⁡6136\log _ { 6 } \frac { 1 } { 36 }log6​361​

Write the equation in exponential form. log⁡17/51=0\log _ { 17 / 5 } 1 = 0log17/5​1=0

Evaluate without the use of a calculator. log⁡t1t\log _ { t } \frac { 1 } { t }logt​t1​

Evaluate without the use of a calculator. log⁡tt19\log _ { t } t ^ { 19 }logt​t19

Write the equation in logarithmic form. 43=644 ^ { 3 } = 6443=64

Write the equation in exponential form. log⁡31729=−6\log _ { 3 } \frac { 1 } { 729 } = - 6log3​7291​=−6

Write the equation in logarithmic form. $5 ^ { - 5 } = \frac { 1 } { 3125 }$

Write the equation in exponential form. $\log _ { 4 } 4096 = 6$

Evaluate without the use of a calculator. $\log 0.001$

Evaluate without the use of a calculator. $\log 10 ^ { 5 }$

Evaluate without the use of a calculator. $\log _ { 8 } \sqrt [ 5 ] { 8 }$

Evaluate without the use of a calculator. $\log \left( 1 \times 10 ^ { 5 } \right)$

Evaluate without the use of a calculator. $\log _ { 2 / 3 } \frac { 8 } { 27 }$

Write the equation in exponential form. $y = \log _ { b } x$

Evaluate without the use of a calculator. $\log _ { 4 } 4$

Evaluate without the use of a calculator. $\log _ { 6 } \frac { 1 } { 36 }$

Write the equation in exponential form. $\log _ { 17 / 5 } 1 = 0$

Evaluate without the use of a calculator. $\log _ { t } \frac { 1 } { t }$

Evaluate without the use of a calculator. $\log _ { t } t ^ { 19 }$

Write the equation in logarithmic form. $4 ^ { 3 } = 64$

Write the equation in exponential form. $\log _ { 3 } \frac { 1 } { 729 } = - 6$