\(\log_2\dfrac18=\log_2 2^{-3}=-3\) and \(\log_5 25=\log_5 5^2=2\). Thus \(-3+2=-1\). Answer: B.
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3
Task
One correct answer
Calculate \(\log_3 18-\log_3 2\).
Solution
Using the logarithm rule, \(\log_3 18-\log_3 2=\log_3\dfrac{18}{2}=\log_3 9=2\). Answer: A.
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4
Task
Numeric answer
Calculate \(\log_a 500-\log_a 4\) if \(\log_5 a=\dfrac14\).
Solution
\(\log_a500-\log_a4=\log_a\dfrac{500}{4}=\log_a125=3\log_a5\). By change of base, \(\log_a5=\dfrac1{\log_5a}=4\). Therefore the value is \(3\cdot4=12\). Answer: \(12\).
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5
Task
Numeric answer
Find the value of \(6^{2\log_6 9-\log_6 4}\).
Solution
\(2\log_6 9=\log_6 81\). Thus the exponent is \(\log_6\dfrac{81}{4}\), so \(6^{\log_6(81/4)}=\dfrac{81}{4}=20.25\). Answer: \(20.25\).
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6
Task
Numeric answer
Calculate \(\log_b a\) if \(\log_3 a=8\) and \(\log_3 b=5\).
Solution
By the change-of-base formula, \(\log_b a=\dfrac{\log_3 a}{\log_3 b}=\dfrac85=1.6\). Answer: \(1.6\).
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7
Task
One correct answer
Calculate \(\log_8 16\).
Solution
Since \(8=2^3\) and \(16=2^4\), \(\log_8 16=\log_{2^3}2^4=\dfrac43\). Answer: B.
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8
Task
One correct answer
If \(\lg b=6\), then \(\lg(10b^2)=\)
Solution
\(\lg(10b^2)=\lg10+\lg b^2=1+2\lg b=1+2\cdot6=13\). Answer: D.
Let \(t=3^x+3^{-x}\). Then \(t^2=9^x+2+9^{-x}=47+2=49\). Since both terms are positive, \(t>0\), so \(t=7\). Answer: A.
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11
Task
One correct answer
Calculate \(36^{\log_6 5}\).
Solution
Since \(36=6^2\), \(36^{\log_6 5}=(6^2)^{\log_6 5}=6^{2\log_6 5}=(6^{\log_6 5})^2=5^2=25\). Answer: D.
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12
Task
Matching
Match each sentence beginning (1–4), where \(a>0\) and \(b>0\), with the correct ending (A–E) to form a true statement.
А
\(a=2b\) Б
\(a=2+b\) В
\(a=4b\) Г
\(a=b^2\) Д
\(a=4+b\)
Solution
1) \(2\log_2 b=\log_2 b^2\), so \(a=b^2\) — D. 2) \(a^3=(2b)^3\), so \(a=2b\) — A. 3) Squaring gives \(a=4b\) — C. 4) \(4\cdot2^b=2^{b+2}\), so \(a=b+2\) — B. Matching: 1–D, 2–A, 3–C, 4–B.
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13
Task
Matching
Positive numbers \(x\) and \(y\) satisfy \(\dfrac{x}{y}=\dfrac14\). Match each expression (1–2) with its value (A–E).
А
\(-2\) Б
\(1.25\) В
\(0.25\) Г
\(2\) Д
\(4\)
Solution
1) \(\dfrac{x+y}{y}=\dfrac{x}{y}+1=\dfrac14+1=\dfrac54=1.25\) — B. 2) \(\log_2x-\log_2y=\log_2\dfrac{x}{y}=\log_2\dfrac14=-2\) — A. Matching: 1–B, 2–A.
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14
Task
One correct answer
Indicate the interval containing \(\log_5 4\).
Solution
Since \(1<4<5\) and the base \(5>1\), \(\log_5 1<\log_5 4<\log_5 5\), so \(0<\log_5 4<1\). Answer: A.
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15
Task
One correct answer
Calculate \(\log_{5^{-2}}5^{\frac12}\).
Solution
Using the common base \(5\), \(\log_{5^{-2}}5^{1/2}=\dfrac{1/2}{-2}=-\dfrac14=-0.25\). Answer: E.
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16
Task
One correct answer
If \(2^a=\dfrac15\), then \(2^{6-a}=\)
Solution
\(2^{6-a}=\dfrac{2^6}{2^a}=\dfrac{64}{1/5}=64\cdot5=320\). Answer: E.
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17
Task
One correct answer
Calculate \(\log_2 5+\log_2 1.6\).
Solution
Using the product rule, \(\log_2 5+\log_2 1.6=\log_2(5\cdot1.6)=\log_2 8=3\). Answer: A.
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18
Task
One correct answer
Calculate \(\log_3 54-\log_3 2\).
Solution
\(\log_3 54-\log_3 2=\log_3\dfrac{54}{2}=\log_3 27=3\). Answer: B.
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19
Task
One correct answer
Indicate the interval containing \(\log_2 9\).
Solution
Since \(8<9<16\), with \(8=2^3\) and \(16=2^4\), we have \(3<\log_2 9<4\). Answer: D.
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20
Task
One correct answer
If \(\log_4 3=a\), then \(\log_{16}9=\)
Solution
\(\log_{16}9=\dfrac{\ln9}{\ln16}=\dfrac{2\ln3}{2\ln4}=\dfrac{\ln3}{\ln4}=\log_4 3=a\). Answer: E.
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21
Task
One correct answer
Calculate \(\log_a\sqrt{ab}\) if \(\log_a b=7\).
Solution
\(\log_a\sqrt{ab}=\dfrac12(\log_a a+\log_a b)=\dfrac12(1+7)=4\). Answer: E.
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22
Task
One correct answer
Calculate \(\log_{\frac1{25}}\sqrt5\).
Solution
\(\dfrac1{25}=5^{-2}\), \(\sqrt5=5^{1/2}\). From \((5^{-2})^x=5^{1/2}\), we get \(-2x=\dfrac12\), so \(x=-\dfrac14\). Answer: A.
Using change of base, the product telescopes: \(\log_3 4\cdot\log_4 5\cdot\log_5 7\cdot\log_7 81=\log_3 81=4\). Answer: \(4\).
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24
Task
One correct answer
Calculate the value of \(\log_3 45+\log_3 900-\log_3 500\).
Solution
\(\log_3 45+\log_3 900-\log_3 500=\log_3\dfrac{45\cdot900}{500}=\log_3 81=4\). Answer: B.
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25
Task
One correct answer
Which of the following intervals contains \(\log_2\dfrac13\)?
Solution
Since \(\dfrac14<\dfrac13<\dfrac12\) and \(y=\log_2x\) is increasing, \(-2<\log_2\dfrac13<-1\). Therefore the number lies in \((-3;-1)\). Answer: B.
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26
Task
Numeric answer
Calculate the value of \(\dfrac{\log_5 27}{\log_5 2-\log_5 162}\).
Solution
\(\log_5 27=3\log_5 3\), while \(\log_5 2-\log_5 162=\log_5\dfrac1{81}=-4\log_5 3\). Hence the value is \(-\dfrac34=-0.75\). Answer: \(-0.75\).
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27
Task
Matching
Match each expression (1–3) with an identically equal expression (A–E), where \(a\) is any negative number.
А
\(0\) Б
\(2a\) В
\(a^2\) Г
\(1\) Д
\(-2a\)
Solution
1) \(a^0=1\) — D. 2) Since \(a<0\), \(|a|=-a\), so \(|a|+a=0\) — A. 3) \(\log_2 2^a=a\), hence \(a\log_2 2^a=a^2\) — C. Matching: 1–D, 2–A, 3–C.
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28
Task
Matching
Match each expression (1–3) with an identically equal expression (A–E), where \(a>0\) and \(a\ne1\).
А
\(a^2\) Б
\(a^7\) В
\(\dfrac1a\) Г
\(a\) Д
\(-a\)
Solution
1) \(a^4:a^3=a\) — D. 2) \(\dfrac{a^2-a}{1-a}=-a\) — E. 3) \(7^{-\log_7 a}=\dfrac1a\) — C. Matching: 1–D, 2–E, 3–C.
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29
Task
Matching
Match each expression (1–3) with the interval (A–E) containing its value if \(a=4.5\).
А
\((-2,0)\) Б
\((0,1)\) В
\((1,2)\) Г
\((2,3)\) Д
\((3,5)\)
Solution
1) \(4.5-2.7=1.8\in(1,2)\) — C. 2) \(\sqrt[3]{3.5-4.5}=-1\in(-2,0)\) — A. 3) Since \(1<4.5<5\), \(0<\log_5 4.5<1\) — B. Matching: 1–C, 2–A, 3–B.
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30
Task
Matching
Match each expression (1–3) with the true statement about its value (A–E) if \(a=-2\dfrac13\).
А
greater than \(5\) Б
belongs to \((0,1)\) В
is negative Г
belongs to \([1,5)\) Д
equals \(0\)
Solution
\(a=-\dfrac73\). 1) \(a^2=\dfrac{49}{9}>5\) — A. 2) Since \(a<0\), \(|a|=-a\), so \(a+|a|=0\) — E. 3) \(\log_5 5^a=a=-\dfrac73<0\) — C. Matching: 1–A, 2–E, 3–C.
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31
Task
One correct answer
Choose the correct double inequality if \(a=0.5^{-1}\), \(b=0.2\), \(c=\log_{0.2}5\).
Solution
\(a=0.5^{-1}=2\), \(b=0.2\). Since \(0.2=\dfrac15\), \(c=\log_{1/5}5=-1\). Thus \(-1\lt0.2\lt2\), so \(c\lt b\lt a\). Answer: A.
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32
Task
Numeric answer
Calculate \(400^{1-\log_{20}4}\).
Solution
Since \(400=20^2\), \(400^{1-\log_{20}4}=20^{2-2\log_{20}4}=\dfrac{20^2}{(20^{\log_{20}4})^2}=\dfrac{400}{16}=25\). Answer: \(25\).
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33
Task
One correct answer
Which row lists \(\log_2 64\), \(\log_{64}2\), and \(11\) in increasing order?
Solution
\(\log_2 64=6\) and \(\log_{64}2=\dfrac16\). Hence \(\dfrac16<6<11\). The increasing order is \(\log_{64}2,\ \log_2 64,\ 11\). Answer: A.
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34
Task
One correct answer
Calculate \(\log_5 49+2\log_5\dfrac57\).
Solution
\(\log_5 49+2\log_5\dfrac57=\log_5\left(49\cdot\dfrac{25}{49}\right)=\log_5 25=2\). Answer: E.
\(\log_{\sqrt3}\sqrt[6]{3}=\dfrac13\), so the first term is \(27^{1/3}=3\). Also \(5^{(\log_5 2)^2}=\left(5^{\log_5 2}\right)^{\log_5 2}=2^{\log_5 2}\), and \(\log_2 16=4\). The last two terms cancel, so the value is \(3\). Answer: C.
