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Quizzes > Mathematics > Algebra

Think You Know Logarithms? Take the Quiz!

Ready for the Hardest Logarithm Questions? Start Your Log Quiz!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art style composition of logarithm symbols and quiz icons on coral background

This logarithm quiz helps you practice hard problems and spot gaps before an exam. You'll work on change of base, expanding logs, solving equations, and tricky roots. After you play, try our practice set or explore more math and logic puzzles .

What is log_10(1000)?
4
2
5
3
The log base 10 of 1000 is the exponent to which 10 must be raised to yield 1000, which is 3. This follows directly from the definition of logarithms. Therefore, log_10(1000) = 3.
What is log_2(16)?
3
2
5
4
The log base 2 of 16 asks for the power to which 2 must be raised to get 16. Since 2^4 = 16, log_2(16) = 4.
What is log_5(1)?
-1
0
1
5
Any nonzero base raised to the power 0 equals 1, so log base 5 of 1 is 0. This property holds for all logarithms: log_b(1) = 0.
What is log_3(27)?
4
3
5
2
Since 3 raised to the 3rd power equals 27, log_3(27) = 3. Logarithms simply invert exponentiation.
What is log_4(1/16)?
2
-4
-2
4
1/16 is 4 raised to the -2 (since 4^2 = 16). Hence log_4(1/16) = -2.
Which property expresses log_b(MN) in terms of log_b(M) and log_b(N)?
log_b(MN) = log_b(M) * log_b(N)
log_b(MN) = log_b(M + N)
log_b(MN) = log_b(M) - log_b(N)
log_b(MN) = log_b(M) + log_b(N)
The product rule for logarithms states that the log of a product is the sum of the logs: log_b(MN) = log_b(M) + log_b(N). This rule follows from exponentiation properties.
Simplify log_7(49).
4
1
2
3
49 is 7 squared, so log_7(49) = 2. Logarithms return the exponent for the base.
Solve the equation log_2(x) = 5 for x.
25
32
64
10
If log_2(x) = 5, then 2^5 = x. Since 2^5 = 32, the solution is x = 32.
Solve log_10(x) = -1 for x.
1
10
-1
0.1
Log base 10 equals -1 means 10^(?1) = x, which is 0.1.
Solve log_3(x+1) = 2 for x.
8
9
7
10
log_3(x+1)=2 implies x+1 = 3^2 = 9, so x = 8.
Simplify log_2(8) + log_2(2).
5
3
2
4
log_2(8)=3 and log_2(2)=1, so their sum is 4.
Using the change-of-base formula, what is log_9(27)?
3/2
2
1/2
1
By change of base, log_9(27) = ln(27)/ln(9). Since 27=3^3 and 9=3^2, this simplifies to (3 ln3)/(2 ln3)=3/2.
Simplify the expression log(a^5) in terms of log(a).
log(a)/5
log(a)^5
5 log(a)
log(a^4)
The power rule for logarithms states log(a^5) = 5 log(a). This rule comes from bringing the exponent out front.
Which of the following represents the change-of-base formula for log_b(a)?
log(b)/log(a)
log(a)/log(b)
b^(log(a))
a^(log(b))
The change-of-base formula states log_b(a) = log(a)/log(b), where the logs on the right can be any common base.
Solve for x: log_2(x+1) + log_2(x-1) = 3.
2
3
4
5
Combine logs: log_2((x+1)(x-1))=3 ? (x+1)(x-1)=8 ? x^2?1=8 ? x^2=9 ? x=±3. Domain requires x>1, so x=3.
Solve for x: log_3(2x-1) = 2.
7
5
4
6
log_3(2x-1)=2 implies 2x?1=3^2=9, so 2x=10 and x=5.
Evaluate log_2(8) * log_8(16).
2
4
6
3
log_2(8)=3 and log_8(16)=4/3, so their product is 3*(4/3)=4.
Simplify log_a(b) * log_b(a).
a+b
ab
0
1
Using change of base, log_a(b)=1/log_b(a). Their product is 1. This is a standard logarithm identity.
Evaluate log_4(32).
2
5/4
5/2
3
4^x=32 ? (2^2)^x=2^(2x)=2^5 ? 2x=5 ? x=5/2. So log_4(32)=5/2.
Simplify ln(ab^2) - ln(a^2 b).
ln(ab)^2
ln(a^2 b^2)
ln(a/b)
ln(b/a)
ln(ab^2)=ln(a)+2 ln(b) and ln(a^2 b)=2 ln(a)+ln(b). Subtracting gives ?ln(a)+ln(b)=ln(b/a).
Solve log(x) + log(x-3) = 1 (base 10).
10
2
-2
5
Combine: log[x(x?3)]=1 ? x^2?3x=10 ? x^2?3x?10=0 ? x=5 or ?2. Domain x>3 gives x=5.
If log_2(x) = a and log_2(y) = b, express log_2(x y^2) in terms of a and b.
2a + b
2a + 2b
a + b
a + 2b
log_2(x y^2)=log_2(x)+log_2(y^2)=a+2 b by properties of logarithms.
Solve log_2(3x) - log_2(x-1) = 2.
8
4
2
5
Combine: log_2(3x/(x?1))=2 ? 3x/(x?1)=4 ? 3x=4x?4 ? x=4. Domain x>1 is satisfied.
Solve log_3(x) + log_3(x-2) = 1.
2
3
1
-1
Combine: log_3[x(x?2)]=1 ? x^2?2x=3 ? x^2?2x?3=0 ? x=3 or ?1. Domain x>2 gives x=3.
Solve log_4(x) + log_8(x) = 5.
8
64
32
16
Convert: log_4(x)=ln(x)/(2 ln2), log_8(x)=ln(x)/(3 ln2). Sum = ln(x)*(5/(6 ln2))=5 ? ln(x)=6 ln2 ? x=2^6=64.
0
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Study Outcomes

  1. Understand fundamental log rules -

    Grasp product, quotient, and power rules to simplify expressions in this free log quiz.

  2. Apply change-of-base formula -

    Use change-of-base techniques to evaluate logarithms in non-standard bases, preparing you for the hardest logarithm question.

  3. Solve diverse logarithmic equations -

    Tackle equations ranging from basic drills to advanced logarithm trivia questions that sharpen your reasoning.

  4. Analyze logarithmic function behaviors -

    Interpret domain, range, and transformations of log functions to enhance your problem-solving skills.

  5. Evaluate and improve your performance -

    Review your answers to identify strengths and weaknesses, guiding further practice in this log quiz online.

Cheat Sheet

  1. Core Logarithm Properties -

    Learn the product, quotient, and power rules to simplify expressions: log_b(MN)=log_bM+log_bN, log_b(M/N)=log_bM−log_bN, and log_b(M^k)=k·log_bM. Mnemonic: "Add for groups, subtract to split, multiply exponents" helps ace any log quiz question. (Source: Khan Academy)

  2. Change of Base Formula -

    Master the formula log_b(a)=log_c(a)/log_c(b) to compute any logarithm using a standard calculator. This trick is handy for log quiz online problems and is often tested in the hardest logarithm question sections. For example, log_2(7)=log₝₀(7)/log₝₀(2). (Source: MIT OpenCourseWare)

  3. Solving Exponential Equations with Logs -

    When variables are in exponents, apply logarithms to isolate them: for 3^x=20, x=ln(20)/ln(3). Practice these step-by-step conversions in a practice logarithm quiz to build confidence. Remember to check for domain restrictions after solving. (Source: University of Cambridge Mathematics)

  4. Understanding Logarithmic Scales -

    Many real-world measures like pH, Richter, and decibel scales are logarithmic, meaning each step represents a tenfold change. Recognizing these contexts can make logarithm trivia questions more intuitive and show the power of logs beyond algebra. Explore applications in a log quiz to see how theory meets practice. (Source: NIST)

  5. Tackling Nested and Composite Logs -

    The hardest logarithm question often involves nested logs like log(log(x)) or combined bases - break them down using core rules. Convert nested expressions step by step and simplify inner logs first, then proceed outward. Strong familiarity with core properties from your log quiz practice will guide you through any composite challenge. (Source: Purplemath)

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