Logarithms

Problem 1

Find $y$ if $\log _{y} \frac{1}{9}=3$ .

See Solution

Problem 2

Find $y$ if $\log_{y} \frac{1}{9} = 3$ .

See Solution

Problem 3

Solve $\log _{2} x + \log _{2}(x-2) = 3$ for $x$ . Options: a) $4,-2$ b) 4 c) -2 d) $2,-4$ e) None.

See Solution

Problem 4

Find the value of $\log _{5} \sqrt{5}$ . Options: a) 2 b) -2 c) $\frac{1}{2}$ d) $\frac{-1}{2}$ e) 0

See Solution

Problem 5

If $4^{x}=40$ , find the value of $x$ : a) $10$ b) $\sqrt[40]{4}$ c) $\sqrt[4]{40}$ d) $\log_{4} 40$ e) $\log_{40} 4$ .

See Solution

Problem 6

Simplify $\log \left(x y^{2}\right)$ . Choose from: a) $2 \log (x y)$ b) $\log (x) \log \left(y^{2}\right)$ c) $2^{x y}$ d) $\log (x)+2 \log (y)$ e) none.

See Solution

Problem 7

If $\log _{a} 64=2$ , find the value of $a$ . Options: a) -8 b) 32 c) 128 d) 4096 e) 8

See Solution

Problem 8

Calculate $\log _{3} 81$ . Choose from: a) 3 b) 9 c) 27 d) 4 e) none of the above.

See Solution

Problem 9

Solve the equation $\log _{3}(x^{2}-9)=1+\log _{3}(3-x)$ .

See Solution

Problem 10

Calculate $\log_2{128}$ .

See Solution

Problem 11

Solve for $x$ in the equation: $x \log _{3} p + 2 \log _{3} p = \log _{3}(p + 1)$ .

See Solution

Problem 12

Encontre $(x, y)$ com $y>1$ para o sistema: $\log_{y}(9x-35)=6$ e $\log_{3y}(27x-81)=3$ .

See Solution

Problem 13

Simplify the expression: $\log 3+\log 7-\log 6$ .

See Solution

Problem 14

Solve for $x$ : $\log _{2}(x+1)=2 \log _{2} 3$ .

See Solution

Problem 15

1. (a) Show that $y^{2}=\frac{1}{2} x^{2}$ if $\log \left(x^{2}+y^{2}\right)-\log 3=\log \left(x^{2}-y^{2}\right)$ . (b) Find $m n$ such that $\log _{3} m+3 \log _{27} n=5$ . (c) If $\ln p=7$ , what is $\log _{p} \sqrt{e}$ ?

See Solution

Problem 16

Solve for $x$ : $3 - \ln(2z - 3) = 2$ .

See Solution

Problem 17

Find the solution set for the inequality $x(-1+\ln x)<0$ .

See Solution

Problem 18

Find $\log_{21} a$ in terms of $p$ and $q$ given that $\log_{a} 3 = p$ and $\log_{a} 7 = q$ .

See Solution

Problem 19

Simplify $\frac{\log _{5} 25 \times \log _{22} 11}{\log _{3} 5}$ without tables or a calculator.

See Solution

Problem 20

Solve for $x$ in these equations: a) $\log _{2} x+\log _{4} x=4.5$ b) $\log (x+2)+\log (x-1)=1$

See Solution

Problem 21

67. (a) Nyatakan syarat-syarat bagi $c$ jika $A=\log _{c} B$ . (b) Dari $\log _{2} y=\frac{5}{\log _{x y} 2}$ , cari $y$ dalam $x$ .

See Solution

Problem 22

Cari nilai $h$ bagi $5 \log _{3}\left(\frac{h}{3}\right)+\log _{3} \sqrt{h}=6$ .

See Solution

Problem 23

Solve for $x$ in the equation $\log 42.117 = 0.4x - 3.08$ .

See Solution

Problem 24

Given $\log_{10} x = a$ , $\log_{10} y = b$ , $\log_{10} z = c$ , express: (i) $10^{2a-3}$ in terms of $x$ ; (ii) $10^{3b-1}$ in terms of $y$ ; (iii) $P = 10^{2a + \frac{b}{2} - 3c}$ in terms of $x, y, z$ ; (iv) $x y$ if $\log_{10} x = a$ , $\log_{10} y = b$ ; (v) $\frac{a^3}{b^2}$ in terms of $\log_{10} a = m$ , $\log_{10} b = n$ ; (vi) $10^{a}$ in terms of $x$ if $\log_{10} x = 2a$ ; (vii) $10^{2b+1}$ in terms of $y$ ; (viii) $P = 10^{3a - 2b}$ in terms of $x, y$ ; (ix) $72^x$ in terms of $y, z$ if $\log_2 y = x$ , $\log_3 z = x$ ; (x) $100^{2a-1}$ in terms of $x, y$ if $\log_2 x = a$ , $\log_5 y = a$ .

See Solution

Problem 25

Solve the equation $\log _{x} 4=1+\log _{2} x$ .

See Solution

Problem 26

Solve the equation: $\frac{1}{2} \ln (x+3)-\ln x=0$ for $x$ .

See Solution

Problem 27

Identify the TRUE statement among these options: A) $\log 8+\log 2=\log 10$ , B) $\log 5 \times \log 3=\log 8$ , C) $\log 9-\log 3=\log 3$ , D) $\log 6-\log 2=\log 4$ .

See Solution

Problem 28

Convert the equation $5^{x}=7$ into logarithmic form.

See Solution

Problem 29

Identify which statements are correct:
I. $(\log_{3} x)^{\frac{1}{3}}=\frac{1}{3} \log_{3} x$
II. $\log_{3}(p q^{5})=\log_{3} p+5 \log_{3} q$
III. $\frac{\log_{7} a}{\log_{7} b}=\log_{7} a-\log_{7} b$

See Solution

Problem 30

Solve the equation $34 \log x - \frac{1}{2} \log (x+4) = 1$ .

See Solution

Problem 31

Solve the equation $\log x - \frac{1}{2} \log (x+4) = 1$ .

See Solution

Problem 32

Evaluate $f(3)$ for $f(x)=x^{2}$ . A. 3 B. 6 C. 9 D. 12 Solve: $3t-2z=4$ , $4t+z=7$ . A. $(2,-1)$ B. $(-1,2)$ C. $(1,3)$ D. $(-3,-2)$ Find the inverse of $f(x)=2$ . A. $f^{-1}(x)=\log 2 x$ B. $f^{-1}(x)=\ln x$ C. $f^{-1}(x)=e^{x}$ D. $f^{-1}(x)=x^{2}$ Calculate $\log_{2} 8$ . A. 2 B. 3 C. 4 D. 8 What is $\log_{5} 25$ ? A. 2 B. 4 C. 6 D. 8 Given $\log_{3} m + \log_{3} m 4 = 2$ , find $m$ . A. 2 B. 4 C. 5 D. 6

See Solution

Problem 33

Convert $2^{x}=3$ to logarithmic form.

See Solution

Problem 34

Identify which statements are correct:
I: $(\log_{2} x)^{\frac{1}{2}}=\frac{1}{2} \log_{2} x$ II: $\frac{\log_{3} x}{\log_{3} y}=\log_{3} x-\log_{3} y$ III: $\log_{5}(x y^{2})=\log_{5} x+2 \log_{5} y$

See Solution

Problem 35

Solve the equation $\log _{9}(2+x)=\log _{16} 4-\log _{\frac{1}{3}} \sqrt{1-2 x}$ .

See Solution

Problem 36

Solve for $x$ in the equation $\log _{9}(2+x)=\log _{16} 4-\log _{1} \sqrt{1-2 x}$ .

See Solution

Problem 37

Solve these equations: (a) $\log _{5} x+1=2 \log _{x} 5$ , (b) $\log _{9}(2+x)=\log _{16} 4-\log _{\frac{1}{3}} \sqrt{1-2 x}$ .

See Solution

Problem 38

Condense and simplify: $2(\log 18 - \log 3) + \frac{1}{2} \log \frac{1}{16} = \log [?]$

See Solution

Problem 39

Find the value of $b$ if the graph of $y=\log _{b} x$ passes through $(8, 2)$ . Choices: a. 10, b. 2, c. $2\sqrt{2}$ , d. $2\sqrt{3}$ .

See Solution

Problem 40

Find the relationship between $x$ and $y$ if the line through points (3,6) and (6,8) represents $Y=\log y$ vs. $X=\log x$ .

See Solution

Problem 41

Find the equivalent equation for $2^{3 x}=10$ . Options include $\log _{2} 10=3 x$ , $\log 2=3 x$ , etc.

See Solution

Problem 42

Solve for $x$ in the equation $\log _{2}(x+4)=3$ .

See Solution

Problem 43

Find the value of $x$ if $\log _{4} 64 = x$ . What is $x$ ?

See Solution

Problem 44

Rewrite the expression as a sum/difference of logarithms: $\log _{2}\left(\frac{x^{4}}{y^{6}}\right)$

See Solution

Problem 45

Solve the equation: $\log_{8} x^{2} = 4$ .

See Solution

Problem 46

Determine the domain of the function $f(x)=\log_{10}\left(\frac{x+2}{x-4}\right)$ .

See Solution

Problem 47

Combine the expression into one logarithm: $4 \log_{c} 9 + 5 \log_{c} 5$ .

See Solution

Problem 48

Find $\log_{b} \frac{A}{B}$ given $\log_{b} A=2$ and $\log_{b} B=-4$ .

See Solution

Problem 49

Find the value of $\log _{4} 1$ .

See Solution

Problem 50

Solve the equation $2^{x+6}=7$ and find the decimal approximation of $x$ to two decimal places.

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Problem 51

Express y in terms of x given that C is a positive constant: $\ln y = \ln 4 x + \ln C$ .

See Solution

Problem 52

Express y in terms of x given: $\ln y = \ln 4 x + \ln C$ where C is a positive constant.

See Solution

Problem 53

Calculate the value of $\log _{5} 125$ .

See Solution

Problem 54

Solve the equation: $\log _{3} x + \log _{3}(x-24) = 4$

See Solution

Problem 55

Rewrite the equation $e^{x}=10$ using logarithms.

See Solution

Problem 56

Evaluate $\log _{3} 25$ using the Change-of-Base Formula and round to two decimal places.

See Solution

Problem 57

Solve the equation: $\log _{3}(x-5)+\log _{3}(x-11)=3$ .

See Solution

Problem 58

Solve the equation: $\ln \sqrt{x+3} = 7$ .

See Solution

Problem 59

Solve for $x$ in the equation: $2+\log _{3}(2x+5)-\log _{3}x=4$ .

See Solution

Problem 60

Find the exact value of $\ln \mathrm{e}^{\sqrt{6}}$ using logarithm properties without a calculator.

See Solution

Problem 61

Rewrite the expression as a sum/difference of logarithms: $\log_{6}\left(\frac{x-1}{x^{4}}\right)$ .

See Solution

Problem 62

Find the domain of the function $f(x)=\log _{3}(x-9)^{2}$ .

See Solution

Problem 63

Find the exact value of $\log _{2} 11 \cdot \log _{11} 8$ using logarithm properties without a calculator.

See Solution

Problem 64

Convert the logarithm $\log _{2} \frac{1}{4}$ to its equivalent exponential form.

See Solution

Problem 65

Find $\log_{b} B^{2}$ if $\log_{b} A = 5$ and $\log_{b} B = -4$ .

See Solution

Problem 66

Solve for $x$ in the equation: $\log _{3}(x+4)=-2$ .

See Solution

Problem 67

Convert the exponential equation $5^{3}=125$ to a logarithmic form.

See Solution

Problem 68

Find $\log_{b} AB$ given that $\log_{b} A=5$ and $\log_{b} B=-2$ .

See Solution

Problem 69

Rewrite the exponential equation $4^{3}=x$ using logarithms.

See Solution

Problem 70

Find the value of $\log _{9} \frac{1}{729}$ .

See Solution

Problem 71

Find the exact value of $7^{\log_{7} 0.499}$ using logarithm properties without a calculator.

See Solution

Problem 72

Rewrite the exponential equation $4^{5 / 2}=32$ using a logarithm.

See Solution

Problem 73

Rewrite the expression $3^{-3}$ using a logarithm.

See Solution

Problem 74

Solve for $x$ in the equation: $e^{x+8}=2$ .

See Solution

Problem 75

Solve the equation: $e^{3 x} = 5$

See Solution

Problem 76

Convert the logarithmic expression to an exponent: $\ln \frac{1}{e^{4}}=-4$ .

See Solution

Problem 77

Find the value of $\ln e^{4}$ .

See Solution

Problem 78

Solve the equation $3^{x+8}=7$ and express the solution using natural logarithms.

See Solution

Problem 79

Calculate the value of $\log _{8} \frac{1}{64}$ .

See Solution

Problem 80

Determine the domain of the function $f(x)=\log (x+6)$ .

See Solution

Problem 81

Find the exact value of $\ln \mathrm{e}$ .

See Solution

Problem 82

Solve for $x$ in the equation: $\log_{2} x = 3$ .

See Solution

Problem 83

Combine the logarithms: $\log_{c} q + \log_{c} r$ as a single logarithm.

See Solution

Problem 84

Find the exact value of $2 \ln e^{4.2}$ using logarithm properties without a calculator.

See Solution

Problem 85

Combine the expression into one logarithm: $2 \log_{c} m - \frac{4}{3} \log_{c} n + \frac{1}{6} \log_{c} j - 6 \log_{c} k$ .

See Solution

Problem 86

Solve the equation $e^{x+7}=5$ and express the solution using natural logarithms.

See Solution

Problem 87

Rewrite the logarithmic expression as an exponent: $\log_{4} x = 3$ .

See Solution

Problem 88

Solve for $x$ in the equation $\log _{4} 64=x$ . Choose from: {16, 3, 256, 68}.

See Solution

Problem 89

Find the value of $x$ in the equation $\log _{4} 64=x$ .

See Solution

Problem 90

Solve the equation $e^{5x} = 3$ and express the solution using natural logarithms.

See Solution

Problem 91

Find the exact value of the expression using logarithm properties: $\log_{3} 6 - \log_{3} 2$ .

See Solution

Problem 92

Find the number of digits in the product of $2^{101}$ and $5^{99}$ .

See Solution

Problem 93

Evaluate the function $g(x, y, z)=\ln (x+y+z)$ for: (a) $g(0,0,0)$ , (b) $g(1,0,0)$ , (c) $g(0,1,0)$ , (d) $g(z, x, y)$ , (e) $g(x+h, y+k, z+I)$ .

See Solution

Problem 94

1. Find the identity element for the operation $a * b = a + b - 3$ . A. 3 B. 2 C. 0 D. -3
2. Calculate the sum of the sequence $4, -2, 1, -\frac{1}{2}, \ldots$ . A. $-\frac{3}{4}$ B. $\frac{3}{4}$ C. $\frac{8}{3}$ D. 8
3. Solve $256^{(x+1)} = 8^{(1-x^{2})}$ . A. $-1, -\frac{5}{3}$ B. $-\frac{3}{8}, -\frac{5}{3}$ C. $\frac{8}{3}, \frac{3}{5}$ D. $\frac{8}{3}, \frac{5}{3}$
4. For roots $\alpha$ and $\beta$ of $x^{2} + 3x - 4 = 0$ , find $\alpha^{2} + \beta^{2} - 3\alpha\beta$ . A. -11 B. 20 C. 21 D. 29
5. Rationalize $\frac{1}{\sqrt{3} - \sqrt{2}}$ . A. $\sqrt{3} - \sqrt{2}$ B. $\frac{\sqrt{3} + \sqrt{2}}{3}$ C. $\frac{\sqrt{3} + \sqrt{2}}{2}$ D. $\sqrt{3} + \sqrt{2}$
6. Solve $\log_{5}(6x + 7) - \log_{5} 6 = 2$ for $x$ .

See Solution

Problem 95

Calculate the $\mathrm{pH}$ for $\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=5.79 \times 10^{-4} \mathrm{M}$ .

See Solution

Problem 96

Calculate the pH for a solution with $\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] = 1.0 \times 10^{-11} M$ .

See Solution

Problem 97

Find the concentration of $\mathrm{H}_{3} \mathrm{O}^{+}$ in a solution with $\mathrm{pH} = 3.6$ using $\mathrm{pH}=-\log_{10}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]$ .

See Solution

Problem 98

Find the concentration of $\mathrm{H}_{3} \mathrm{O}^{+}$ in moles per liter for a solution with a $\mathrm{pH}$ of $9.000$ .

See Solution

Problem 99

Which expression equals $\log \frac{B^{A}}{D^{C}}$ ? F. $A C \log (B-D)$ G. $A \log B-C \log D$ H. $B \log A+D \log C$ J. $B \log A-D \log C$ K. $A \log B \div C \log D$

See Solution

Problem 100

Find the value of $\ln e^{8}$ .

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111

Logarithms

Problem 1

Find y y y if log⁡y19=3 \log _{y} \frac{1}{9}=3 logy​91​=3.

Problem 2

Find y y y if log⁡y19=3 \log_{y} \frac{1}{9} = 3 logy​91​=3.

Problem 3

Solve log⁡2x+log⁡2(x−2)=3\log _{2} x + \log _{2}(x-2) = 3log2​x+log2​(x−2)=3 for xxx. Options: a) 4,−24,-24,−2 b) 4 c) -2 d) 2,−42,-42,−4 e) None.

Problem 4

Find the value of log⁡55\log _{5} \sqrt{5}log5​5​. Options: a) 2 b) -2 c) 12\frac{1}{2}21​ d) −12\frac{-1}{2}2−1​ e) 0

Problem 5

If 4x=404^{x}=404x=40, find the value of xxx: a) 101010 b) 440\sqrt[40]{4}404​ c) 404\sqrt[4]{40}440​ d) log⁡440\log_{4} 40log4​40 e) log⁡404\log_{40} 4log40​4.

Problem 6

Simplify log⁡(xy2)\log \left(x y^{2}\right)log(xy2). Choose from: a) 2log⁡(xy)2 \log (x y)2log(xy) b) log⁡(x)log⁡(y2)\log (x) \log \left(y^{2}\right)log(x)log(y2) c) 2xy2^{x y}2xy d) log⁡(x)+2log⁡(y)\log (x)+2 \log (y)log(x)+2log(y) e) none.

Problem 7

If log⁡a64=2\log _{a} 64=2loga​64=2, find the value of aaa. Options: a) -8 b) 32 c) 128 d) 4096 e) 8

Problem 8

Calculate log⁡381\log _{3} 81log3​81. Choose from: a) 3 b) 9 c) 27 d) 4 e) none of the above.

Problem 9

Solve the equation log⁡3(x2−9)=1+log⁡3(3−x)\log _{3}(x^{2}-9)=1+\log _{3}(3-x)log3​(x2−9)=1+log3​(3−x).

Problem 10

Calculate log⁡2128\log_2{128}log2​128.

Problem 11

Solve for xxx in the equation: xlog⁡3p+2log⁡3p=log⁡3(p+1)x \log _{3} p + 2 \log _{3} p = \log _{3}(p + 1)xlog3​p+2log3​p=log3​(p+1).

Problem 12

Encontre (x,y)(x, y)(x,y) com y>1y>1y>1 para o sistema: log⁡y(9x−35)=6\log_{y}(9x-35)=6logy​(9x−35)=6 e log⁡3y(27x−81)=3\log_{3y}(27x-81)=3log3y​(27x−81)=3.

Problem 13

Simplify the expression: log⁡3+log⁡7−log⁡6\log 3+\log 7-\log 6log3+log7−log6.

Problem 14

Solve for xxx: log⁡2(x+1)=2log⁡23\log _{2}(x+1)=2 \log _{2} 3log2​(x+1)=2log2​3.

Problem 15

Problem 16

Solve for xxx: 3−ln⁡(2z−3)=23 - \ln(2z - 3) = 23−ln(2z−3)=2.

Problem 17

Find the solution set for the inequality x(−1+ln⁡x)<0x(-1+\ln x)<0x(−1+lnx)<0.

Problem 18

Find log⁡21a\log_{21} alog21​a in terms of ppp and qqq given that log⁡a3=p\log_{a} 3 = ploga​3=p and log⁡a7=q\log_{a} 7 = qloga​7=q.

Problem 19

Simplify log⁡525×log⁡2211log⁡35\frac{\log _{5} 25 \times \log _{22} 11}{\log _{3} 5}log3​5log5​25×log22​11​ without tables or a calculator.

Problem 20

Solve for xxx in these equations: a) log⁡2x+log⁡4x=4.5\log _{2} x+\log _{4} x=4.5log2​x+log4​x=4.5 b) log⁡(x+2)+log⁡(x−1)=1\log (x+2)+\log (x-1)=1log(x+2)+log(x−1)=1

Problem 21

67. (a) Nyatakan syarat-syarat bagi ccc jika A=log⁡cBA=\log _{c} BA=logc​B. (b) Dari log⁡2y=5log⁡xy2\log _{2} y=\frac{5}{\log _{x y} 2}log2​y=logxy​25​, cari yyy dalam xxx.

Problem 22

Cari nilai hhh bagi 5log⁡3(h3)+log⁡3h=65 \log _{3}\left(\frac{h}{3}\right)+\log _{3} \sqrt{h}=65log3​(3h​)+log3​h​=6.

Problem 23

Solve for xxx in the equation log⁡42.117=0.4x−3.08\log 42.117 = 0.4x - 3.08log42.117=0.4x−3.08.

Problem 24

Problem 25

Solve the equation log⁡x4=1+log⁡2x\log _{x} 4=1+\log _{2} xlogx​4=1+log2​x.

Problem 26

Solve the equation: 12ln⁡(x+3)−ln⁡x=0\frac{1}{2} \ln (x+3)-\ln x=021​ln(x+3)−lnx=0 for xxx.

Problem 27

Problem 28

Convert the equation 5x=75^{x}=75x=7 into logarithmic form.

Problem 29

Problem 30

Solve the equation 34log⁡x−12log⁡(x+4)=134 \log x - \frac{1}{2} \log (x+4) = 134logx−21​log(x+4)=1.

Problem 31

Solve the equation log⁡x−12log⁡(x+4)=1\log x - \frac{1}{2} \log (x+4) = 1logx−21​log(x+4)=1.

Problem 32

Problem 33

Convert 2x=32^{x}=32x=3 to logarithmic form.

Problem 34

Problem 35

Solve the equation log⁡9(2+x)=log⁡164−log⁡131−2x\log _{9}(2+x)=\log _{16} 4-\log _{\frac{1}{3}} \sqrt{1-2 x}log9​(2+x)=log16​4−log31​​1−2x​.

Problem 36

Solve for xxx in the equation log⁡9(2+x)=log⁡164−log⁡11−2x\log _{9}(2+x)=\log _{16} 4-\log _{1} \sqrt{1-2 x}log9​(2+x)=log16​4−log1​1−2x​.

Problem 37

Solve these equations: (a) log⁡5x+1=2log⁡x5\log _{5} x+1=2 \log _{x} 5log5​x+1=2logx​5, (b) log⁡9(2+x)=log⁡164−log⁡131−2x\log _{9}(2+x)=\log _{16} 4-\log _{\frac{1}{3}} \sqrt{1-2 x}log9​(2+x)=log16​4−log31​​1−2x​.

Problem 38

Condense and simplify: 2(log⁡18−log⁡3)+12log⁡116=log⁡[?]2(\log 18 - \log 3) + \frac{1}{2} \log \frac{1}{16} = \log [?]2(log18−log3)+21​log161​=log[?]

Problem 39

Find the value of bbb if the graph of y=log⁡bxy=\log _{b} xy=logb​x passes through (8,2)(8, 2)(8,2). Choices: a. 10, b. 2, c. 222\sqrt{2}22​, d. 232\sqrt{3}23​.

Problem 40

Find the relationship between xxx and yyy if the line through points (3,6) and (6,8) represents Y=log⁡yY=\log yY=logy vs. X=log⁡xX=\log xX=logx.

Problem 41

Find the equivalent equation for 23x=102^{3 x}=1023x=10. Options include log⁡210=3x\log _{2} 10=3 xlog2​10=3x, log⁡2=3x\log 2=3 xlog2=3x, etc.

Problem 42

Solve for xxx in the equation log⁡2(x+4)=3\log _{2}(x+4)=3log2​(x+4)=3.

Problem 43

Find $y$ if $\log _{y} \frac{1}{9}=3$ .

Find $y$ if $\log_{y} \frac{1}{9} = 3$ .

Solve $\log _{2} x + \log _{2}(x-2) = 3$ for $x$ . Options: a) $4,-2$ b) 4 c) -2 d) $2,-4$ e) None.

Find the value of $\log _{5} \sqrt{5}$ . Options: a) 2 b) -2 c) $\frac{1}{2}$ d) $\frac{-1}{2}$ e) 0

If $4^{x}=40$ , find the value of $x$ : a) $10$ b) $\sqrt[40]{4}$ c) $\sqrt[4]{40}$ d) $\log_{4} 40$ e) $\log_{40} 4$ .

Simplify $\log \left(x y^{2}\right)$ . Choose from: a) $2 \log (x y)$ b) $\log (x) \log \left(y^{2}\right)$ c) $2^{x y}$ d) $\log (x)+2 \log (y)$ e) none.

If $\log _{a} 64=2$ , find the value of $a$ . Options: a) -8 b) 32 c) 128 d) 4096 e) 8

Calculate $\log _{3} 81$ . Choose from: a) 3 b) 9 c) 27 d) 4 e) none of the above.

Solve the equation $\log _{3}(x^{2}-9)=1+\log _{3}(3-x)$ .

Calculate $\log_2{128}$ .

Solve for $x$ in the equation: $x \log _{3} p + 2 \log _{3} p = \log _{3}(p + 1)$ .

Encontre $(x, y)$ com $y>1$ para o sistema: $\log_{y}(9x-35)=6$ e $\log_{3y}(27x-81)=3$ .

Simplify the expression: $\log 3+\log 7-\log 6$ .

Solve for $x$ : $\log _{2}(x+1)=2 \log _{2} 3$ .

Solve for $x$ : $3 - \ln(2z - 3) = 2$ .

Find the solution set for the inequality $x(-1+\ln x)<0$ .

Find $\log_{21} a$ in terms of $p$ and $q$ given that $\log_{a} 3 = p$ and $\log_{a} 7 = q$ .

Simplify $\frac{\log _{5} 25 \times \log _{22} 11}{\log _{3} 5}$ without tables or a calculator.

Solve for $x$ in these equations: a) $\log _{2} x+\log _{4} x=4.5$ b) $\log (x+2)+\log (x-1)=1$

67. (a) Nyatakan syarat-syarat bagi $c$ jika $A=\log _{c} B$ . (b) Dari $\log _{2} y=\frac{5}{\log _{x y} 2}$ , cari $y$ dalam $x$ .

Cari nilai $h$ bagi $5 \log _{3}\left(\frac{h}{3}\right)+\log _{3} \sqrt{h}=6$ .

Solve for $x$ in the equation $\log 42.117 = 0.4x - 3.08$ .

Solve the equation $\log _{x} 4=1+\log _{2} x$ .

Solve the equation: $\frac{1}{2} \ln (x+3)-\ln x=0$ for $x$ .

Convert the equation $5^{x}=7$ into logarithmic form.

Solve the equation $34 \log x - \frac{1}{2} \log (x+4) = 1$ .

Solve the equation $\log x - \frac{1}{2} \log (x+4) = 1$ .

Convert $2^{x}=3$ to logarithmic form.

Solve the equation $\log _{9}(2+x)=\log _{16} 4-\log _{\frac{1}{3}} \sqrt{1-2 x}$ .

Solve for $x$ in the equation $\log _{9}(2+x)=\log _{16} 4-\log _{1} \sqrt{1-2 x}$ .

Solve these equations: (a) $\log _{5} x+1=2 \log _{x} 5$ , (b) $\log _{9}(2+x)=\log _{16} 4-\log _{\frac{1}{3}} \sqrt{1-2 x}$ .

Condense and simplify: $2(\log 18 - \log 3) + \frac{1}{2} \log \frac{1}{16} = \log [?]$

Find the value of $b$ if the graph of $y=\log _{b} x$ passes through $(8, 2)$ . Choices: a. 10, b. 2, c. $2\sqrt{2}$ , d. $2\sqrt{3}$ .

Find the relationship between $x$ and $y$ if the line through points (3,6) and (6,8) represents $Y=\log y$ vs. $X=\log x$ .

Find the equivalent equation for $2^{3 x}=10$ . Options include $\log _{2} 10=3 x$ , $\log 2=3 x$ , etc.

Solve for $x$ in the equation $\log _{2}(x+4)=3$ .

Find the value of $x$ if $\log _{4} 64 = x$ . What is $x$ ?

Rewrite the expression as a sum/difference of logarithms: $\log _{2}\left(\frac{x^{4}}{y^{6}}\right)$

Solve the equation: $\log_{8} x^{2} = 4$ .

Determine the domain of the function $f(x)=\log_{10}\left(\frac{x+2}{x-4}\right)$ .

Combine the expression into one logarithm: $4 \log_{c} 9 + 5 \log_{c} 5$ .

Find $\log_{b} \frac{A}{B}$ given $\log_{b} A=2$ and $\log_{b} B=-4$ .

Find the value of $\log _{4} 1$ .

Solve the equation $2^{x+6}=7$ and find the decimal approximation of $x$ to two decimal places.

Express y in terms of x given that C is a positive constant: $\ln y = \ln 4 x + \ln C$ .

Express y in terms of x given: $\ln y = \ln 4 x + \ln C$ where C is a positive constant.

Calculate the value of $\log _{5} 125$ .

Solve the equation: $\log _{3} x + \log _{3}(x-24) = 4$

Rewrite the equation $e^{x}=10$ using logarithms.

Evaluate $\log _{3} 25$ using the Change-of-Base Formula and round to two decimal places.

Solve the equation: $\log _{3}(x-5)+\log _{3}(x-11)=3$ .

Solve the equation: $\ln \sqrt{x+3} = 7$ .

Solve for $x$ in the equation: $2+\log _{3}(2x+5)-\log _{3}x=4$ .

Find the exact value of $\ln \mathrm{e}^{\sqrt{6}}$ using logarithm properties without a calculator.

Rewrite the expression as a sum/difference of logarithms: $\log_{6}\left(\frac{x-1}{x^{4}}\right)$ .

Find the domain of the function $f(x)=\log _{3}(x-9)^{2}$ .

Find the exact value of $\log _{2} 11 \cdot \log _{11} 8$ using logarithm properties without a calculator.

Convert the logarithm $\log _{2} \frac{1}{4}$ to its equivalent exponential form.

Find $\log_{b} B^{2}$ if $\log_{b} A = 5$ and $\log_{b} B = -4$ .

Solve for $x$ in the equation: $\log _{3}(x+4)=-2$ .

Convert the exponential equation $5^{3}=125$ to a logarithmic form.

Find $\log_{b} AB$ given that $\log_{b} A=5$ and $\log_{b} B=-2$ .

Rewrite the exponential equation $4^{3}=x$ using logarithms.

Find the value of $\log _{9} \frac{1}{729}$ .

Find the exact value of $7^{\log_{7} 0.499}$ using logarithm properties without a calculator.

Rewrite the exponential equation $4^{5 / 2}=32$ using a logarithm.

Rewrite the expression $3^{-3}$ using a logarithm.

Solve for $x$ in the equation: $e^{x+8}=2$ .