Let’s solve each logarithm step by step. Remember:
logₐ(b) = c means aᶜ = b
We’ll go one by one.
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9) log₃81
What power of 3 gives 81?
3¹ = 3
3² = 9
3³ = 27
3⁴ = 81 → So,
4
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10) log₄(1/2)
What power of 4 gives 1/2?
Write 4 as 2², and 1/2 as 2⁻¹
So: (2²)^x = 2⁻¹ → 2^(2x) = 2⁻¹ → 2x = -1 → x =
-1/2
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11) log₁₂144
12² = 144 → So,
2
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12) log₆(1/36)
36 = 6², so 1/36 = 6⁻² → So,
-2
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13) log_(2/3)(9/4)
Note: 9/4 = (3/2)² = [(2/3)⁻¹]² = (2/3)⁻²
So, exponent is
-2
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14) log₀.₂₅4
0.25 = 1/4 = 4⁻¹
So we want: (4⁻¹)^x = 4 → 4^(-x) = 4¹ → -x = 1 → x =
-1
Wait — let’s double-check:
Is (0.25)^(-1) = 4? Yes! Because 1/(0.25) = 4 → So answer is
-1
Actually, wait — if base is 0.25 and result is 4, then:
Let x = log₀.₂₅4 → 0.25ˣ = 4
But 0.25 = 1/4 → (1/4)ˣ = 4 → 4⁻ˣ = 4¹ → -x = 1 → x =
-1
Yes, correct.
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15) log₃(-3)
Logarithms are only defined for positive numbers. You can’t take log of negative number →
Undefined
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16) log₈4
8 = 2³, 4 = 2²
So: (2³)^x = 2² → 2^(3x) = 2² → 3x = 2 → x =
2/3
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17) log₈₁(1/27)
81 = 3⁴, 27 = 3³ → 1/27 = 3⁻³
So: (3⁴)^x = 3⁻³ → 3^(4x) = 3⁻³ → 4x = -3 → x =
-3/4
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18) log_(1/16)32
1/16 = 2⁻⁴, 32 = 2⁵
So: (2⁻⁴)^x = 2⁵ → 2^(-4x) = 2⁵ → -4x = 5 → x =
-5/4
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19) log₄0
Log of zero is undefined →
Undefined
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20) log₁₀1
Any base to the 0 power is 1 → 10⁰ = 1 → So,
0
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21) log₄(1/8)
4 = 2², 8 = 2³ → 1/8 = 2⁻³
So: (2²)^x = 2⁻³ → 2^(2x) = 2⁻³ → 2x = -3 → x =
-3/2
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22) log₂₇(1/3)
27 = 3³, 1/3 = 3⁻¹
So: (3³)^x = 3⁻¹ → 3^(3x) = 3⁻¹ → 3x = -1 → x =
-1/3
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23) log₉3
9 = 3², so (3²)^x = 3¹ → 3^(2x) = 3¹ → 2x = 1 → x =
1/2
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24) log₆(6³ˣ)
This is just the exponent:
3x (because logₐ(a) = k)
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25) log₃₆(1/6)
36 = 6², 1/6 = 6⁻¹
So: (6²)^x = 6⁻¹ → 6^(2x) = 6⁻¹ → 2x = -1 → x =
-1/2
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26) log₁₂₈2
128 = 2⁷ → So: (2⁷)^x = 2¹ → 2^(7x) = 2¹ → 7x = 1 → x =
1/7
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27) log_(1/4)16
1/4 = 4⁻¹, 16 = 4²
So: (4⁻¹)^x = 4² → 4^(-x) = 4² → -x = 2 → x =
-2
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28) log_z(z²ˣ)
Again, logₐ(a) = k → So,
2x
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29) ln(e¹²)
ln is log base e → logₑ(e¹²) =
12
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30) 3^(log₃5)
This is inverse: a^(logₐb) = b → So,
5
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31) ln1
e⁰ = 1 → So,
0
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32) e^(ln4x)
Again, inverse: e^(ln y) = y → So,
4x
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Final Answer:
9) 4
10) -1/2
11) 2
12) -2
13) -2
14) -1
15) Undefined
16) 2/3
17) -3/4
18) -5/4
19) Undefined
20) 0
21) -3/2
22) -1/3
23) 1/2
24) 3x
25) -1/2
26) 1/7
27) -2
28) 2x
29) 12
30) 5
31) 0
32) 4x
Parent Tip: Review the logic above to help your child master the concept of logarithm problems worksheet.
