Unlock hundreds more features
Save your Quiz to the Dashboard
View and Export Results
Use AI to Create Quizzes and Analyse Results

OR
Sign inSign in with Facebook
Sign inSign in with Google
Quizzes > High School Quizzes > Mathematics

Expanding Logarithms Practice Quiz

Sharpen skills with step-by-step logarithm problems

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art depicting a trivia quiz on mastering log expansions for algebra students.

Expand log(AB).
log A + log B
log A - log B
log (A+B)
log A * log B
Using the product rule for logarithms, log(AB) expands to log A + log B. This property is fundamental in logarithmic expansions.
Expand log(A/B).
log A - log B
log A + log B
log (A-B)
log (A)/log (B)
The quotient rule for logarithms tells us that log(A/B) equals log A minus log B. This is a direct application of logarithm properties.
Expand log(x^4).
4 log x
log 4x
log x + 4
log (4x)
By the power rule, the exponent 4 can be brought in front of the logarithm as a multiplier, resulting in 4 log x. This rule simplifies expressions with exponents.
Expand log(2xy).
log2 + log x + log y
2 log(xy)
log2 + log x - log y
log2xy
Using the product rule, log(2xy) is expanded into the sum log2 + log x + log y. Each factor inside the logarithm is separated accordingly.
Expand ln(ez).
1 + ln z
ln e + z
ln e * ln z
e + ln z
Since ln(e) is equal to 1, using the product rule gives ln(ez) = ln e + ln z = 1 + ln z. This is a straightforward example of logarithm expansion.
Expand log(a^2 * b / c^3) fully in terms of log a, log b, and log c.
2 log a + log b - 3 log c
log a^2 + log b + log c^3
2 log a + log b + 3 log c
log a + 2 log b - 3 log c
Apply the power rule to write log(a^2) as 2 log a and log(c^3) as 3 log c. The quotient rule then yields the final expansion: 2 log a + log b - 3 log c.
Expand log[(xy^2)/(z^3)].
log x + 2 log y - 3 log z
2 log x + log y - 3 log z
log x + log y - log z
2 log x + 2 log y - log z
Using the power rule on y^2 gives 2 log y and on z^3 gives 3 log z. Then the product and quotient rules combine to yield log x + 2 log y - 3 log z.
Which expression represents the correct expansion of log(4x^3y^2)?
log4 + 3 log x + 2 log y
4 log x + x log 3y^2
3 log4 + log x + 2 log y
log 4x + log 3y + log2
Separate the constant and variable parts using the product rule, and then apply the power rule to the variables with exponents. The correct expansion is log4 + 3 log x + 2 log y.
Expand ln((2x^2)/(3y)).
ln2 + 2 ln x - ln3 - ln y
2 ln x + ln y - ln2 - ln3
ln2 + ln x - ln3y
ln(2x^2) - ln(3y^2)
Apply the quotient rule to write ln((2x^2)/(3y)) as ln(2x^2) - ln(3y). Then use the power rule on x^2 to get 2 ln x, resulting in ln2 + 2 ln x - ln3 - ln y.
Which of the following is the expansion for log[(m^3 n)/(p^2 q)]?
3 log m + log n - 2 log p - log q
log m^3 + log n - log p^2 - log q
3 log m + 2 log n - log p - log q
3 log m + log n - log p - log q
By applying the power, product, and quotient rules, the expression expands to 3 log m + log n - 2 log p - log q. Each exponent is brought in front as a multiplier.
Expand ln[(5ab^4)/c^2] into a sum of logarithms.
ln5 + ln a + 4 ln b - 2 ln c
5 ln a + 4 ln b - c ln2
ln(5a) + ln(b^4) - ln(c^2)
ln5 + 4 ln a + ln b - 2 ln c
First, split the numerator into ln5 + ln a + ln(b^4) and then apply the power rule to convert ln(b^4) into 4 ln b. The denominator ln(c^2) becomes 2 ln c, which is subtracted from the sum.
Expand log[(2x)/(5y^3)].
log2 + log x - log5 - 3 log y
log2 + log x - 3 log5 - log y
2 log x - 5 log y
log2 + log5 + log x - 3 log y
Using the product and quotient rules, separate the numerator and denominator. The exponent 3 on y becomes 3 log y via the power rule, resulting in log2 + log x - log5 - 3 log y.
What is the correct expansion of log[(7x^2y^3)/(2z)]?
log7 + 2 log x + 3 log y - log2 - log z
7 log x + 2 log y + 3 log y - 2 log z
log7x^2y^3 - log2z
log7 + log x^2 + log y^3 - log2z
First, apply the power rule to the components x^2 and y^3. Then, using the product rule for the numerator and the quotient rule for division by 2z, the full expansion is obtained.
Expand ln[10x/(y^2z)].
ln10 + ln x - 2 ln y - ln z
ln10 + ln x - ln y - ln z
ln(10x) - ln(y^2z)
ln10x - ln2y - ln z
Break down the expression using the quotient rule and then apply the power rule on y^2 to get 2 ln y. The expansion becomes ln10 + ln x - 2 ln y - ln z.
Determine the expanded form of log[(a b^2)^(1/2)].
1/2 log a + log b
log a + 1/2 log b
log a + log b
(log a + log b)/2
Taking the square root of (a b^2) is equivalent to raising it to the 1/2 power. Apply the power rule to obtain 1/2[log a + log b^2] which simplifies to 1/2 log a + log b.
Expand log[(2a^3b)^(1/3)/(5c^2)] fully.
1/3 log2 + log a + 1/3 log b - log5 - 2 log c
1/3 log2 + 3 log a + 1/3 log b - log5 - 2 log c
log2 + log a + log b - log5 - 2 log c
1/3 log2 + log a + 1/3 log b - log5 - log c
Begin by applying the power rule to the numerator: (2a^3b)^(1/3) becomes 2^(1/3) a b^(1/3). Then use the quotient rule to subtract log(5c^2), which expands to log5 + 2 log c.
Fully expand ln[(x^(3/2) * y^(1/3))/(z^(5/2) * w)].
(3/2) ln x + (1/3) ln y - (5/2) ln z - ln w
ln x^(3/2) + ln y^(1/3) - ln z^(5/2) - ln w
3 ln x + ln y - 5 ln z - ln w
1/2 ln x + 1/3 ln y - 5/2 ln z - ln w
Apply the power rule to each term: ln(x^(3/2)) becomes (3/2) ln x and ln(y^(1/3)) becomes (1/3) ln y; similarly, ln(z^(5/2)) is (5/2) ln z. Then use the quotient rule to subtract the logarithms of the denominator.
Expand log[ (4x^3y^2)^(2) / (8z^4)^(1/2) ].
log16 + 6 log x + 4 log y - (1/2) log8 - 2 log z
log16 + 6 log x + 4 log y - log8 - 2 log z
log16 + 6 log x + 4 log y - (1/2) log8 - log z
log16 + 6 log x + 4 log y - 2 log z
First, raise (4x^3y^2) to the power of 2 to obtain 16x^6y^4, and take the square root of (8z^4) to get (8z^4)^(1/2) = (1/2)[log8 + 4 log z]. Subtracting yields the expansion: log16 + 6 log x + 4 log y - (1/2) log8 - 2 log z.
Expand ln[((a^2b)^3)/(c^(1/2)d^4)] fully.
6 ln a + 3 ln b - 1/2 ln c - 4 ln d
2 ln a + 3 ln b - 1/2 ln c - 4 ln d
6 ln a + 3 ln b - ln c - 4 ln d
6 ln a + ln b - 1/2 ln c - 4 ln d
The numerator (a^2b)^3 simplifies to a^6b^3. Applying the power rule gives 6 ln a and 3 ln b. The denominator contributes 1/2 ln c and 4 ln d via the power rule, and the quotient rule sets the latter as subtracted terms.
Expand log[(25x^2√y)/(8z^(3/2))].
log25 + 2 log x + 1/2 log y - log8 - 3/2 log z
log25 + 2 log x + log y - log8 - 3/2 log z
log25 + 2 log x + 1/2 log y - 3 log8 - 3/2 log z
log25 + log x + 1/2 log y - log8 - 3/2 log z
Rewrite √y as y^(1/2) and z^(3/2) remains with its fractional exponent. Then apply the product and power rules to get log25 + 2 log x + 1/2 log y for the numerator and subtract log8 + 3/2 log z from the denominator.
0
{"name":"Expand log(AB).", "url":"https://www.quiz-maker.com/QPREVIEW","txt":"Expand log(AB)., Expand log(A\/B)., Expand log(x^4).","img":"https://www.quiz-maker.com/3012/images/ogquiz.png"}

Study Outcomes

  1. Apply logarithmic expansion rules to rewrite complex logarithmic expressions.
  2. Analyze the properties of logarithms, including product, quotient, and power rules.
  3. Simplify logarithmic equations using expansion techniques.
  4. Verify the correctness of expanded forms through reverse calculations.
  5. Solve algebraic problems involving logarithmic expansions for test preparation.

Expanding Logarithms Practice Cheat Sheet

  1. Master the Product Rule - When you see a product inside a logarithm, you can split it into separate logs and simply add them. For example, log₝(xy) = log₝(x) + log₝(y). This trick turns tough multiplication problems into easy addition exercises! GeeksforGeeks Guide
  2. Understand the Quotient Rule - Logs turn division into subtraction: log₝(x/y) = log₝(x) - log₝(y). This rule helps you break apart fractions and simplify them step by step. Practice with different numerators and denominators to see how subtraction replaces division! GeeksforGeeks Guide
  3. Apply the Power Rule - Pull exponents out front: log₝(x❿) = n · log₝(x). It's like having a "power lever" that makes calculations a breeze. Use this rule to handle big exponents by converting them into a simple multiplier! GeeksforGeeks Guide
  4. Learn the Change of Base Formula - Want a log in an uncommon base? Use log₝(x) = log_c(x) ÷ log_c(a) with any new base c (often 10 or e). This formula is your gateway to calculators and tables that only handle base‑10 or natural logs. It's a universal translator for logarithms! GeeksforGeeks Guide
  5. Recognize the Zero Rule - The log of 1 is always zero, no matter the base: log₝(1) = 0. Since any number to the zero power equals 1, logs of one collapse to zero. Keep this in your toolkit as a quick check on tricky problems! GeeksforGeeks Guide
  6. Understand the Identity Rule - When the argument and the base match, the log gives you one: log₝(a) = 1. It's the mirror moment in logs where the input equals the base. Use it to spot shortcuts and confirm your steps! GeeksforGeeks Guide
  7. Practice Expanding Logarithms - Break down complex log expressions with product, quotient, and power rules to simplify them. The more you expand, the more patterns you'll spot, and the faster you'll solve. Try challenging exercises to flex your expansion muscles! ChiliMath Lesson
  8. Condense Logarithmic Expressions - Combine multiple logs into one neat package by reversing the expansion rules. This skill is essential for solving equations and proving identities. Practice on mixed sums and differences to master the art of condensation! OpenStax Textbook
  9. Solve Logarithmic Equations - Use your log properties to isolate variables and crack equations like log₝(x) + 2 = 5 or log₝(x²) = 3. Turning exponentials into linear forms makes this process smoother. Work through examples step by step to build confidence! OpenStax Textbook
  10. Practice with Real-World Applications - Logs pop up everywhere: measuring acidity (pH), modeling sound intensity (decibels), and tracking population growth. Seeing logs in action helps cement the concepts and keeps studying fun. Explore these applications to see logs come alive! OpenStax Textbook
Make a Quiz (FREE) Copy & Edit This Quiz Create a Quiz with AI

Mathematics Quizzes

Powered by: Quiz Maker