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Quizzes > Mathematics > Algebra

Ready to Master Condensing and Expanding Logs? Take the Quiz!

Think you can ace expanding logs practice? Dive into our expanding condensing logarithms quiz now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration with log expressions and math symbols on coral background for expanding and condensing logs quiz

Are you ready to put your condensing and expanding logs skills to the test? This interactive quiz on condensing and expanding logs invites learners to challenge their expanding condensing logarithms knowledge, strengthen expanding logs practice, and master expanding logarithms practice in a fun, engaging way. Whether you're brushing up on key techniques or exploring advanced twists like expanding logs with square roots , you'll find each question sharpens your critical thinking. Dive into our quick logarithm quiz , track your progress, and see how you measure up. Ready to elevate your skills? Let's begin!

Expand log(2x^3) using log properties.
log2 + 3 logx
log2 + logx^3
3 log2 + logx
log(2x) + log3
Using the product rule, log(ab)=log a+log b, and the power rule, log(a^n)=n log a, we get log(2x^3)=log2+log(x^3)=log2+3 logx. This breaks the original log into separate, simpler terms. For more on these properties, see this page.
Condense log x + log y into a single logarithm.
log(x/y)
log(xy)
log(x+y)
log x + y
The product rule for logarithms states that log a + log b = log(ab). Therefore log x + log y condenses to log(xy). This is a basic and common log identity. More examples can be found here.
Expand ln(?(xy)) into separate logarithms.
½ ln x + ½ ln y
ln?x + ln?y
ln x + ln y
2 ln x + 2 ln y
Since ?(xy) = (xy)^(1/2), ln((xy)^(1/2)) = (1/2) ln(xy), and by the product rule that equals ½ ln x + ½ ln y. This uses both the power rule and the product rule. A detailed walkthrough is available here.
Condense ln x ? 2 ln y into a single logarithmic expression.
ln(x) ? ln(2y)
ln(x²/y)
ln(x/y²)
ln(xy²)
The subtraction rule ln a ? ln b = ln(a/b) and the power rule ln(b^n) = n ln b give ln x ? 2 ln y = ln x ? ln(y²) = ln(x/y²). This simplifies the expression into one log. More on condensing logs is found here.
Expand log?((x?y)/z) fully.
log? x + log? y ? log? z
log? x + 2 log? y ? log? z
log?(x?y) ? log? z
log? x + ½ log? y ? log? z
First apply the quotient rule: log?(x?y) ? log? z. Then ?y = y^(1/2) gives log? x + log?(y^(1/2)) ? log? z = log? x + ½ log? y ? log? z. See more at this resource.
Condense log a + log b ? log c ? log d into one logarithm.
log((a+b)/(c+d))
log(ab) ? log(cd)
log(ab/(cd))
log((ab)(cd))
Combine additions: log a + log b = log(ab), and subtractions: ?log c ? log d = ?log(cd), so total is log(ab) ? log(cd) = log(ab/(cd)). This uses both product and quotient rules. For more details, visit Purplemath.
Expand log??((2?x)/(3y²)) into a sum and difference of logs.
log?? 2 + ½ log?? x ? log?? 3 ? 2 log?? y
log?? 2 + log?? x ? log?? 3 ? log?? y
2 log?? x ? log?? 3 + log?? y
log? 2 + ½ log?? x ? log?? 3 ? 2 log?? y
Apply quotient rule then product and power rules: log(2?x) ? log(3y²) = [log2 + ½ log x] ? [log3 + 2 log y] = log2 + ½ log x ? log3 ? 2 log y. Detailed steps are at Khan Academy.
Condense ½ ln(x+1) + ln(x?1) ? ³/? ln(x²) into a single logarithm.
ln((x+1)½+(x?1)?(x²)³/?)
ln([?(x+1)(x?1)]/x)
ln((x+1)(x?1)/x³)
ln([?(x+1)(x?1)]/x³)
Use each coefficient as a power: ½ ln(x+1)=ln((x+1)^(½)), ln(x?1)=ln(x?1), ³/? ln(x²)=ln((x²)^(³/?))=ln(x³). Then combine: ln(?(x+1)*(x?1))?ln(x³)=ln([?(x+1)(x?1)]/x³). See rules at Purplemath.
Expand ln((e^x·?y)/(z³)) completely.
x + ½ ln y ? 3 ln z
ln(e^x) + ln y ? ln(z³)
e^x + ln y ? 3 ln z
x + ln y^(1/2) ? ln z³
Break the quotient: ln(e^x·?y) ? ln(z³). Then ln(e^x)=x, ln(?y)=½ ln y, ln(z³)=3 ln z. Sum: x + ½ ln y ? 3 ln z. More at this link.
Condense 3 log x + 2 log(x?1) ? log(x+2) into one logarithm.
log([x(x?1)²]/(x+2)³)
log(x³ + (x?1)² ? (x+2))
log(x³(x?1)²(x+2))
log([x³(x?1)²]/(x+2))
Apply the power rule: 3 log x=log(x³), 2 log(x?1)=log((x?1)²). Then subtract log(x+2) gives log((x³(x?1)²)/(x+2)). Learn more here.
Condense ½ ln(x²?1) ? ln(x?1) into a single natural logarithm.
ln(?((x+1)/(x?1)))
ln((x+1)/(x?1))
ln((x²?1)/(x?1))
½ ln((x+1)/(x?1))
Since x²?1=(x?1)(x+1), ½ ln(x²?1)=½[ln(x?1)+ln(x+1)]. Subtract ln(x?1) gives ½ ln(x+1)?½ ln(x?1)=ln(?((x+1)/(x?1))). For an in-depth guide see this reference.
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Study Outcomes

  1. Expand Logarithmic Expressions -

    Use product, quotient, and power rules to decompose complex logarithms into sums and differences of simpler logs.

  2. Condense Logarithmic Terms -

    Apply inverse properties and exponent rules to combine multiple log terms into a single, concise expression.

  3. Apply Logarithm Properties -

    Select and implement the correct log property for each problem to simplify or restructure expressions effectively.

  4. Identify Common Pitfalls -

    Recognize and avoid typical mistakes in condensing and expanding logs through targeted quiz questions and instant feedback.

  5. Evaluate and Correct Solutions -

    Assess your own answers to reinforce understanding and use feedback to master expanding logs practice.

  6. Build Speed and Confidence -

    Sharpen your expanding and condensing logarithms skills under time-based scenarios to improve fluency and accuracy.

Cheat Sheet

  1. Fundamental Logarithm Rules -

    Master the product, quotient, and power rules: log_b(xy)=log_b x + log_b y, log_b(x/y)=log_b x − log_b y, and log_b(x^k)=k·log_b x. These core identities form the building blocks for condensing and expanding logs and are backed by resources like Khan Academy and MIT OpenCourseWare. A handy mnemonic is "PoQuP" for Product, Quotient, Power to recall the order of operations.

  2. Condensing Logarithmic Expressions -

    Combine sums and differences of logs into a single expression, for example log_a M + 2 log_a N − log_a P = log_a(M·N²/P). This technique, highlighted in Stewart's Calculus texts, streamlines complex expressions in condensing and expanding logs practice. Try condensing varied problems to build speed and accuracy.

  3. Expanding Complex Logarithms -

    Expand a single logarithm into sums and differences, such as log_2((x³√y)/z²) = 3 log_2 x + ½ log_2 y − 2 log_2 z. This approach, emphasized by University of California math resources, helps you dissect nested exponents and radicals. Regular expanding logarithms practice solidifies pattern recognition and manipulation skills.

  4. Change-of-Base Formula -

    Use log_b A = log_k A ∕ log_k b to switch any logarithm to base k (commonly 10 or e) when your calculator has limited bases. This versatile formula, featured in MIT OpenCourseWare, broadens problem-solving flexibility. Challenge yourself with base-conversion exercises to enhance your expanding logs practice.

  5. Domain Considerations and Common Pitfalls -

    Always ensure the argument of a log is positive and the base is positive and not equal to 1 to avoid invalid operations. This critical check, stressed in academic sources like Stewart's Calculus, prevents domain errors in condensing and expanding logs. Frame a quick mental checklist - x>0 and b>0, b≠1 - before each manipulation.

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