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Quizzes > High School Quizzes > Mathematics

Log Properties Practice Quiz

Sharpen Log Skills Through Engaging Quiz Challenges

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art promoting a trivia quiz on mastering log properties for high school algebra students

Easy
Which of the following best describes the equation log_b(a) = c?
c^b = a
b^a = c
b^c = a
a^c = b
The definition of a logarithm states that log_b(a) = c if and only if b raised to the power c equals a. This fundamental relationship bridges exponentials and logarithms.
Convert the exponential equation 10^4 = 10000 into logarithmic form.
log_10(4) = 10000
log_10000(4) = 10
log_10(10000) = 4
log_10000(10) = 4
Converting an exponential equation to logarithmic form involves interchanging the roles of the base and the exponent. Since 10^4 equals 10000, the logarithmic form is log_10(10000) = 4.
Simplify the expression log(5) + log(2).
log(3)
log(7)
log(10)
2 log(10)
Using the product rule for logarithms, log(5) + log(2) can be combined as log(5Ã - 2) which is log(10). This property simplifies the sum of logarithms with the same base.
Which logarithmic property allows you to rewrite k log(a) as log(a^k)?
Quotient Rule
Product Rule
Power Rule
Change of Base Rule
The power rule for logarithms permits moving a coefficient in front of a log into the exponent of the argument. This rule is essential when simplifying logarithmic expressions.
Simplify the expression log(a) - log(b).
log(ab)
log(b/a)
log(a/b)
log(a) / log(b)
The quotient rule of logarithms states that the difference log(a) - log(b) is equivalent to log(a/b). This property helps combine or simplify logarithmic expressions.
Medium
Solve for x: log(x) = 3 (base 10).
10
30
1000
3
Since logarithms are the inverses of exponentials, log(x) = 3 implies that 10 raised to the power of 3 equals x. Therefore, x = 10^3, which is 1000.
Apply the quotient rule: What is log(9) - log(3) equal to?
log(6)
log(27)
log(3)
log(1)
By the quotient rule, the difference log(9) - log(3) simplifies to log(9/3). Since 9 divided by 3 is 3, the expression equals log(3).
Simplify 2 log(7) using logarithmic properties.
log(7) - log(2)
log(49)
log(343)
log(14)
Using the power rule, 2 log(7) is equivalent to log(7^2). Since 7^2 equals 49, the expression simplifies to log(49).
Which of the following equals log_2(16)?
2
4
16
8
Because 2 raised to the power of 4 equals 16, log_2(16) is 4. This demonstrates how logarithms serve as the inverse of exponential functions.
Express log_5(25) using the change of base formula.
25/log(5)
log(5)/log(25)
log_25(5)
log(25)/log(5)
The change of base formula states that log_b(a) can be expressed as log(a)/log(b) for any chosen logarithm base. Thus, log_5(25) becomes log(25)/log(5).
If log_b(64) = 6, what is the value of b?
2
8
4
16
The equation log_b(64) = 6 means that b^6 = 64. Recognizing that 64 can be written as 2^6 allows us to determine that b must be 2.
Which property of logarithms lets you combine log(3) + log(5) into a single logarithm?
Change of Base Rule
Quotient Rule
Product Rule
Power Rule
The product rule for logarithms indicates that the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments. Hence, log(3) + log(5) equals log(15).
Simplify the expression 3 log(x) - log(y).
log(3x^3) - log(y)
log(3x/y)
log(x/y^3)
log(x^3/y)
First, apply the power rule to rewrite 3 log(x) as log(x^3). Then, using the quotient rule, log(x^3) - log(y) combines into log(x^3/y).
Rewrite log_8(64) in exponential form and determine its value.
4
8
2
3
The logarithm log_8(64) = c means that 8^c = 64. Expressing both numbers as powers of 2 (8 = 2^3 and 64 = 2^6) shows that 3c = 6, so c = 2.
Evaluate log_4(32) using the change of base formula.
4
2
5/2
3
Using the change of base formula, log_4(32) can be rewritten as log_2(32)/log_2(4). Since 32 = 2^5 and 4 = 2^2, this simplifies to 5/2.
Hard
Solve for x: log_3(2x - 4) = 2.
7
6
8
6.5
Rewriting the logarithmic equation in exponential form gives 2x - 4 = 3^2, which is 9. Solving 2x - 4 = 9 results in x = 6.5. It is crucial to perform the algebraic steps carefully.
Solve the equation: log_2(x) + log_2(x - 2) = 3.
6
8
4
2
Combining the logarithms using the product rule yields log_2[x(x - 2)] = 3, which means x(x - 2) = 2^3 or 8. Solving the quadratic equation leads to x = 4 as the only valid solution considering the domain restrictions.
Determine x given that log(x) + log(2x) = 1, where log denotes the common logarithm.
5
2.5
10
√5
Using the product rule, log(x) + log(2x) becomes log(2x^2) = 1. This implies that 2x^2 = 10, so x^2 = 5. Since x must be positive, the solution is x = √5.
Solve the equation: log_5(x) + log_5(x - 4) = 1.
5
1
-5
4
By combining the logarithms with the product rule, the equation becomes log_5[x(x - 4)] = 1, which indicates that x(x - 4) = 5. Solving the resulting quadratic equation and considering the domain yields x = 5 as the valid solution.
If log_a(b) = 1/2 and log_b(a) = 2, what is the relationship between a and b?
a = 2b
b = a^2
b = 2a
a = b^2
From log_a(b) = 1/2, we deduce that b = a^(1/2), and from log_b(a) = 2, it follows that a = b^2. These relationships are consistent, confirming that a = b^2. This problem tests the understanding of inverse logarithmic relationships.
0
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Study Outcomes

  1. Understand the fundamental definitions of logarithms and their notations.
  2. Apply logarithmic properties to simplify complex expressions.
  3. Analyze and solve logarithmic equations efficiently.
  4. Evaluate the effects of logarithmic operations on different algebraic structures.
  5. Synthesize problem-solving strategies to tackle real exam-style questions involving logarithms.

Log Properties Practice Cheat Sheet

  1. Product Rule - Break up products inside a log into a sum on the outside for super-speedy simplification. For example, loga(xy) = loga(x) + loga(y), which means you can tackle big expressions in bite-sized chunks. Learn the Product Rule
  2. Quotient Rule - Split a log of a fraction into a subtraction problem that's easy to handle. With loga(x/y) = loga(x) - loga(y), you'll breeze through division under the log. Master the Quotient Rule
  3. Power Rule - Pull exponents out front so you can multiply instead of wrestling with powers inside the log. That means loga(xn) = n · loga(x), giving you a turbo boost when simplifying. Discover the Power Rule
  4. Change of Base Formula - Swap bases like a pro, even when your calculator doesn't cooperate. Use loga(b) = logc(b) / logc(a) to switch to any base you like. Explore Change of Base
  5. Expanding Logarithmic Expressions - Turn complex logs into a sum or difference of simpler logs. For example, log2(12) becomes log2(4) + log2(3), so you can see each piece clearly. Practice Expanding Logs
  6. Condensing Logarithmic Expressions - Squash multiple log terms into one neat expression for cleaner answers. For instance, log3(x) + log3(y) condenses to log3(xy), saving you space and time. Try Condensing Logs
  7. Solving Logarithmic Equations - Convert logs into exponent form or vice versa to find the mystery variable. If log2(x) = 3, you instantly see that x = 23 = 8. Solve Log Equations
  8. Inverse Relationship - Remember that logs and exponents are two sides of the same coin. If ax = b, then loga(b) = x, making conversions a cinch. Understand Inverses
  9. Common & Natural Logs - Get cozy with base-10 logs (log) and base-e logs (ln). Knowing when to use log10(x) versus ln(x) can speed up calculator work and exam success. Explore Common & Natural Logs
  10. Graphing Logarithmic Functions - Visualize domains, ranges, and key points to ace graph problems. For example, y = log2(x) only exists for x > 0 and stretches infinitely up and down. Graph Log Functions
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