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NY Regents Algebra II Properties of Logarithms — practice questions

Explore 4 free Properties of Logarithms questions in NY Regents Algebra II Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Properties of Logarithms questions and exercises with answers

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The logarithm of a product rewrites as which combination of logarithms?Read the answer and worked solution →Can log of the quantity x squared plus y squared be expanded into separate logarithms?Read the answer and worked solution →Expand log base 3 of (9 times x to the fifth) as far as possible.Read the answer and worked solution →Fill the three laws of logarithmsRead the answer and worked solution →

Worked practice questions

The logarithm of a product rewrites as which combination of logarithms?

  1. A product of logarithms
  2. A quotient of logarithms
  3. A power of logarithms
  4. A sum of logarithms
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Correct answer: D. A sum of logarithms

1. Logarithms are exponents, and multiplying like bases adds exponents. 2. So the logarithm of a product equals a sum of logarithms. 3. The rule can be applied again and again, so a product of four factors becomes four separate logarithms added. _Source: OpenStax Precalculus (CC BY 4.0), section 4.5 Logarithmic Properties_

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Can log of the quantity x squared plus y squared be expanded into separate logarithms?

  1. Yes, it becomes 2 log x plus 2 log y
  2. Yes, it becomes log x plus log y
  3. No, unless x and y are positive
  4. No, sums inside cannot be split
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Correct answer: D. No, sums inside cannot be split

1. The product, quotient and power rules cover multiplication, division and exponents only. 2. Addition inside the argument matches none of them. 3. So a logarithm of a sum cannot be expanded into separate logarithms. 4. Splitting it into log x plus log y would silently swap the sum for a product. _Source: OpenStax Precalculus (CC BY 4.0), section 4.5 Logarithmic Properties_

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Expand log base 3 of (9 times x to the fifth) as far as possible.

  1. 2 + 5 log base 3 of x
  2. 9 + 5 log base 3 of x
  3. 2 times 5 log base 3 of x
  4. 5 + 2 log base 3 of x
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Correct answer: A. 2 + 5 log base 3 of x

1. The argument is a product, so the product rule splits it into two logarithms added. 2. That gives log base 3 of 9 plus log base 3 of x to the fifth. 3. The power rule moves the exponent 5 in front of the second logarithm. 4. That turns the second piece into 5 times log base 3 of x. 5. The first piece can be evaluated, since 3 squared is 9, so log base 3 of 9 is 2. 6. Putting the pieces together gives 2 plus 5 log base 3 of x. 7. Answering 9 plus 5 log base 3 of x leaves the first logarithm unevaluated and drops its logarithm sign. 8. Answering 2 times 5 log base 3 of x uses a product where the product rule demands a sum. 9. Answering 5 plus 2 log base 3 of x swaps the two numbers, applying the exponent to the wrong factor. _Source: OpenStax Precalculus (CC BY 4.0), section 4.5 Logarithmic Properties_

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More Mathematics topics

Complex NumbersExponential and Logarithmic ModelsGraphs of Polynomial FunctionsPower and Polynomial FunctionsRational FunctionsZeros of Polynomial FunctionsQuadratic FunctionsDividing Polynomials