Question 1

Assume that x,y,and z are positive numbers.Use the properties of logarithms to write the expression $- 2 \log _ { b } x - 6 \log _ { b } y + \frac { 1 } { 7 } \log _ { b } z$ as the logarithm of one quantity. ?
A) $\log _ { b } \frac { z ^ { 1 / 7 } } { x ^ { 6 } y ^ { 2 } }$
B) $\log _ { b } \frac { z ^ { 1 / 6 } } { x ^ { 2 } y ^ { 7 } }$
C) $\log _ { b } \frac { z ^ { 1 / 2 } } { x ^ { 7 } y ^ { 6 } }$
D) $\log _ { b } \frac { x ^ { 1 / 7 } } { y ^ { 2 } z ^ { 6 } }$
E) $\log _ { b } \frac { z ^ { 1 / 7 } } { x ^ { 2 } y ^ { 6 } }$

Accepted Answer

The expression can be rewritten using the power rule of logarithms: $-2 \log_b x = \log_b (x^{-2})$, $-6 \log_b y = \log_b (y^{-6})$, and $\frac{1}{7} \log_b z = \log_b (z^{1/7})$. Combining these gives $\log_b \left(\frac{z^{1/7}}{x^2 y^6}\right)$.

Question 2

Assume that x is a positive number.Use the properties of logarithms to write the expression ? Log_b (x + 8)- log_b x as the logarithm of one quantity. ? A) $\log _ { b } \frac { x ^ { 2 } + 8 } { x }$ B) $\log _ { b } \frac { x + 8 } { x }$ C) $\log _ { b } \frac { x - 8 } { x }$ D) $\log _ { b } \frac { x ^ { 2 } + 8 } { 8 }$ E)log_b (x² - 8x)

Accepted Answer

The properties of logarithms state that $\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)$. Applying this to the given expression, we get $\log_b\left(\frac{x + 8}{x}\right)$.

Question 3

Condense the expression 7(log x - log y)to the logarithm of a single term.
A) $\log \left( \frac { x } { y } \right) ^ { 7 }$
B) $\log \frac { 7 x } { 7 y }$
C) $\log \frac { x ^ { 7 } } { y }$
D) $\log \frac { x ^ { 7 } } { \sqrt [ 7 ] { y } }$
E)7(log x - log y)

Accepted Answer

The expression can be simplified using the logarithm properties: $7(\log x - \log y) = \log \left( \frac { x } { y } \right) ^ { 7 }$.

Question 4

Assume that x,y,and c are positive numbers.Use the properties of logarithms to write the expression $\log _ { c } 5 x y$ in terms of the logarithms of x and y. ?
A) $\log _ { c } 5 + \log _ { c } x y$
B) $\log _ { c } 5 + \log _ { c } x + \log _ { c } y$
C) $\log _ { c } 5 + \log _ { c } x$
D) $\log _ { c } 5 + \log _ { 5 } x + \log _ { 5 } y$
E) $\log _ { c } 5 + \log _ { c } y$

Accepted Answer

The properties of logarithms allow us to separate the multiplication inside a log into the sum of logs. Thus, $\log_c(5xy) = \log_c(5) + \log_c(x) + \log_c(y)$.

Question 5

Simplify the expression $\log _ { 3 } \left( \frac { 1 } { 27 } \right) ^ { 3 }$ .
A)3
B)0
C)-9
D)-81
E)The expression cannot be simplified.

Accepted Answer

The expression simplifies to $\log_3(1/27)^3 = \log_3(1/27^3) = \log_3(1/19683) = \log_3(3^{-9}) = -9$, using the property $\log_b(a^n) = n\log_b(a)$ and knowing that $27 = 3^3$, so $1/27^3 = 3^{-9}$.

Question 6

Condense the expression $$\frac { 1 } { 3 } \left[ \log _ { 6 } x + \log _ { 6 } 7 \right] - \left[ \log _ { 6 } y \right]$$ to the logarithm of a single term.
A) $\log _ { 6 } \frac { ( 7 x ) ^ { 3 } } { y }$
B) $\log _ { 6 } \frac { \sqrt [ 3 ] { 7 x } } { y }$
C) $\log _ { 6 } \sqrt [ 3 ] { \frac { 7 x } { y } }$
D) $\log _ { 6 } \sqrt [ 3 ] { 7 x } - \log _ { 6 } y$
E) $\log _ { 6 } \frac { 7 x } { 3 y }$

Accepted Answer

The answer of Condense the expression \[\frac { 1 }...

Question 7

Condense the expression 3(log x - log y)to the logarithm of a single term.
A) $\log \frac { x ^ { 3 } } { y }$
B) $\log \frac { 3 x } { 3 y }$
C) $\log \frac { x } { y ^ { 3 } }$
D)3(log x - log y)
E) $\log \left( \frac { x } { y } \right) ^ { 3 }$

Accepted Answer

Using the logarithm properties, $3(\log x - \log y)$ simplifies to $\log \left( \frac { x } { y } \right) ^ { 3 }$. This is because the difference of logs is the log of a quotient, and multiplying a log by a constant is equivalent to raising its argument to the power of that constant.

Question 8

Find the exact value of $\ln e ^ { 250 } - \ln \sqrt { e }$ without using a calculator.
A)5
B)2.5
C)1.25
D)2
E)3

Accepted Answer

The answer of Find the exact value of \(\ln e...

Question 9

Assume that x,y,and c are positive numbers.Use the properties of logarithms to write the expression $\log _ { c } \sqrt [ 6 ] { x y }$ in terms of the logarithms of x and y. ?
A) $\frac { 1 } { 6 } \log _ { c } x + \log _ { c } y$
B) $6 \log _ { c } x + 6 \log _ { c } y$
C) $\frac { 1 } { 6 } \log _ { c } ( x + y )$
D) $\log _ { c } x + \log _ { c } y$
E) $\frac { 1 } { 6 } \log _ { c } x + \frac { 1 } { 6 } \log _ { c } y$

Accepted Answer

The expression $\log _ { c } \sqrt [ 6 ] { x y }$ can be simplified using the properties of logarithms. First, the sixth root can be expressed as a power of $1/6$, and then the product $xy$ inside the logarithm allows the use of the product rule for logarithms. This results in $\frac { 1 } { 6 } \log _ { c } x + \frac { 1 } { 6 } \log _ { c } y$.

Question 10

Assume that x,y and c are positive numbers.Use the properties of logarithms to write the expression $\log _ { c } x ^ { 6 } y ^ { 4 }$ in terms of the logarithms of x and y. ?
A) $6 \log _ { c } x + 4 \log _ { c } y$
B) $24 \log _ { c } x + y$
C) $6 \log _ { c } x + 6 \log _ { c } y$
D) $24 \log _ { c } x + 4 \log _ { c } y$
E) $6 \log _ { c } x + 24 \log _ { c } y$

Accepted Answer

Using the properties of logarithms, specifically the power rule ($\log_b(m^n) = n \log_b(m)$) and the product rule ($\log_b(mn) = \log_b(m) + \log_b(n)$), the expression $\log_c(x^6y^4)$ can be rewritten as $6\log_c(x) + 4\log_c(y)$.

Question 11

Assume that x,y,z and b are positive numbers.Use the properties of logarithms to write the expression $\log _ { b } \sqrt [ 4 ] { \frac { x ^ { 5 } y ^ { 2 } } { z ^ { 4 } } }$ in terms of the logarithms of x,y,and z. ?
A) $\frac { 5 } { 4 } \log _ { b } x + \log _ { b } y - \log _ { b } z$
B) $\frac { 5 } { 4 } \log _ { b } x + \frac { 1 } { 2 } \log _ { b } y - \log _ { b } z$
C) $\log _ { b } x + \frac { 1 } { 2 } \log _ { b } y - \log _ { b } z$
D) $\frac { 5 } { 4 } \log _ { b } ( x + y + z )$
E) $20 \log _ { b } x + 8 \log _ { b } y - 16 \log _ { b } z$

Accepted Answer

The expression can be simplified using the properties of logarithms: the logarithm of a root is equal to the logarithm of the radicand divided by the index of the root, and the logarithm of a quotient is the difference of the logarithms. Thus, $\log_b \sqrt[4]{\frac{x^5y^2}{z^4}}$ simplifies to $\frac{5}{4}\log_b x + \frac{1}{2}\log_b y - \log_b z$.

Question 12

Put the expressions in the appropriate order: $\frac { \ln 8 } { \ln e } , \frac { \ln 8 } { e } , \ln 8$ .
A) $\ln 8 < \frac { \ln 8 } { e } < \frac { \ln 8 } { \ln e }$
B) $\frac { \ln 8 } { e } < \frac { \ln 8 } { \ln e } = \ln 8$
C)? $\ln 8 < \frac { \ln 8 } { e } = \frac { \ln 8 } { \ln e }$
D) $\frac { \ln 8 } { \ln e } < \ln 8 < \frac { \ln 8 } { \ln e }$
E)The expressions are equivalent.

Accepted Answer

The answer of Put the expressions in the appropriate order:...

Question 13

Condense the expression $\frac { 1 } { 5 } \left[ \log _ { 4 } x + \log _ { 4 } 3 \right] - \left[ \log _ { 4 } y \right]$ to the logarithm of a single term.
A) $\log _ { 4 } \frac { ( 3 x ) ^ { 5 } } { y }$
B) $\log _ { 4 } \frac { 3 x } { 5 y }$
C) $\log _ { 4 } \sqrt [ 5 ] { \frac { 3 x } { y } }$
D) $\log _ { 4 } \frac { \sqrt [ 5 ] { 3 x } } { y }$
E) $\log _ { 4 } \sqrt [ 5 ] { 3 x } - \log _ { 4 } y$

Accepted Answer

The expression can be condensed using the properties of logarithms: $\frac{1}{5}[\log_4 x + \log_4 3] = \log_4((3x)^{1/5})$. Subtracting $\log_4 y$ gives $\log_4 \frac{\sqrt[5]{3x}}{y}$.

Question 14

Find the exact value of $\log _ { 7 } \sqrt [ 3 ] { 49 }$ without using a calculator.
A) $\frac { 49 } { 3 }$
B) $\frac { 3 } { 49 }$
C)? $\frac { 14 } { 3 }$
D) $\frac { 2 } { 3 }$
E)-1

Accepted Answer

The expression $\log _ { 7 } \sqrt [ 3 ] { 49 }$ simplifies to $\log _ { 7 } 7^{2/3}$, which equals $\frac { 2 } { 3 }$ because the base and the argument have the same base (7), making the exponent the value of the logarithm.

Question 15

The pH of an acidic solution is a measure of the concentration of the acid particles in the solution,with smaller values of the pH indicating higher acid concentration.In a laboratory experiment,the pH of a certain acid solution is changed by dissolving over-the-counter antacid tablets into the solution.In this experiment,the pH changes according to the equation $\mathrm { pH } = 5.14 + \log \left( \frac { x } { 0.40 - x } \right)$ , where x is the number of grams of antacid added to the solution.What is the pH of the solution after the addition of 0.05 grams of antacid tablet?
A)5.99
B)4.02
C)-0.85
D)4.29
E)5.14

Accepted Answer

The answer of The pH of an acidic solution is...

Question 16

Condense the expression log₄ x + log₄ 3 to the logarithm of a single term. A)log₄ -3x B)log₄ 3x C)log₄ x³ D)log(3x)⁴ E)log₄ (x + 3)

Accepted Answer

Use the property of logarithms that says log(a) + log(b) = log(ab) to combine the two expressions into one logarithm: log₄ x + log₄ 3 = log₄ (x)(3) = log₄ 3x

Question 17

Find the exact value of $\log _ { 4 } 44 - \log _ { 4 } 11$ without using a calculator.
A)1
B)4
C)11
D) $\frac { 11 } { 2 }$
E) $\frac { 1 } { 2 }$

Accepted Answer

Using the logarithm property $\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)$, we get $\log_4(44) - \log_4(11) = \log_4\left(\frac{44}{11}\right) = \log_4(4) = 1$.

Quiz 20: Properties of Logarithms

Assume that x,y,and z are positive numbers.Use the properties of logarithms to write the expression $- 2 \log _ { b } x - 6 \log _ { b } y + \frac { 1 } { 7 } \log _ { b } z$ as the logarithm of one quantity. ?

Assume that x is a positive number.Use the properties of logarithms to write the expression ? Log_b (x + 8) - log_b x as the logarithm of one quantity. ?

Condense the expression 7(log x - log y) to the logarithm of a single term.

Assume that x,y,and c are positive numbers.Use the properties of logarithms to write the expression $\log _ { c } 5 x y$ in terms of the logarithms of x and y. ?

Simplify the expression $\log _ { 3 } \left( \frac { 1 } { 27 } \right) ^ { 3 }$ .

Condense the expression $\frac { 1 } { 3 } \left[ \log _ { 6 } x + \log _ { 6 } 7 \right] - \left[ \log _ { 6 } y \right]$ to the logarithm of a single term.

Condense the expression 3(log x - log y) to the logarithm of a single term.

Find the exact value of $\ln e ^ { 250 } - \ln \sqrt { e }$ without using a calculator.

Assume that x,y,and c are positive numbers.Use the properties of logarithms to write the expression $\log _ { c } \sqrt [ 6 ] { x y }$ in terms of the logarithms of x and y. ?

Assume that x,y and c are positive numbers.Use the properties of logarithms to write the expression $\log _ { c } x ^ { 6 } y ^ { 4 }$ in terms of the logarithms of x and y. ?

Assume that x,y,z and b are positive numbers.Use the properties of logarithms to write the expression $\log _ { b } \sqrt [ 4 ] { \frac { x ^ { 5 } y ^ { 2 } } { z ^ { 4 } } }$ in terms of the logarithms of x,y,and z. ?

Put the expressions in the appropriate order: $\frac { \ln 8 } { \ln e } , \frac { \ln 8 } { e } , \ln 8$ .

Condense the expression $\frac { 1 } { 5 } \left[ \log _ { 4 } x + \log _ { 4 } 3 \right] - \left[ \log _ { 4 } y \right]$ to the logarithm of a single term.

Find the exact value of $\log _ { 7 } \sqrt [ 3 ] { 49 }$ without using a calculator.

Condense the expression log₄ x + log₄ 3 to the logarithm of a single term.

Find the exact value of $\log _ { 4 } 44 - \log _ { 4 } 11$ without using a calculator.

Quiz 20: Properties of Logarithms

Assume that x,y,and z are positive numbers.Use the properties of logarithms to write the expression −2log⁡bx−6log⁡by+17log⁡bz- 2 \log _ { b } x - 6 \log _ { b } y + \frac { 1 } { 7 } \log _ { b } z−2logb​x−6logb​y+71​logb​z as the logarithm of one quantity. ?

Assume that x is a positive number.Use the properties of logarithms to write the expression ? Logb (x + 8) - logb x as the logarithm of one quantity. ?

Condense the expression 7(log x - log y) to the logarithm of a single term.

Assume that x,y,and c are positive numbers.Use the properties of logarithms to write the expression log⁡c5xy\log _ { c } 5 x ylogc​5xy in terms of the logarithms of x and y. ?

Simplify the expression log⁡3(127) 3\log _ { 3 } \left( \frac { 1 } { 27 } \right) ^ { 3 }log3​(271​) 3 .

Condense the expression 13[log⁡6x+log⁡67]−[log⁡6y]\frac { 1 } { 3 } \left[ \log _ { 6 } x + \log _ { 6 } 7 \right] - \left[ \log _ { 6 } y \right]31​[log6​x+log6​7]−[log6​y] to the logarithm of a single term.

Condense the expression 3(log x - log y) to the logarithm of a single term.

Find the exact value of ln⁡e250−ln⁡e\ln e ^ { 250 } - \ln \sqrt { e }lne250−lne​ without using a calculator.

Assume that x,y,and c are positive numbers.Use the properties of logarithms to write the expression log⁡cxy6\log _ { c } \sqrt [ 6 ] { x y }logc​6xy​ in terms of the logarithms of x and y. ?

Assume that x,y and c are positive numbers.Use the properties of logarithms to write the expression log⁡cx6y4\log _ { c } x ^ { 6 } y ^ { 4 }logc​x6y4 in terms of the logarithms of x and y. ?

Assume that x,y,z and b are positive numbers.Use the properties of logarithms to write the expression log⁡bx5y2z44\log _ { b } \sqrt [ 4 ] { \frac { x ^ { 5 } y ^ { 2 } } { z ^ { 4 } } }logb​4z4x5y2​​ in terms of the logarithms of x,y,and z. ?

Put the expressions in the appropriate order: ln⁡8ln⁡e,ln⁡8e,ln⁡8\frac { \ln 8 } { \ln e } , \frac { \ln 8 } { e } , \ln 8lneln8​,eln8​,ln8 .

Condense the expression 15[log⁡4x+log⁡43]−[log⁡4y]\frac { 1 } { 5 } \left[ \log _ { 4 } x + \log _ { 4 } 3 \right] - \left[ \log _ { 4 } y \right]51​[log4​x+log4​3]−[log4​y] to the logarithm of a single term.

Find the exact value of log⁡7493\log _ { 7 } \sqrt [ 3 ] { 49 }log7​349​ without using a calculator.

Condense the expression log4 x + log4 3 to the logarithm of a single term.

Find the exact value of log⁡444−log⁡411\log _ { 4 } 44 - \log _ { 4 } 11log4​44−log4​11 without using a calculator.