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Topics · Logarithms

Logarithms in DSE Maths Paper 2

Logarithms let you work with powers by working with their exponents. Questions ask you to use the laws of logarithms and the change of base, solve equations involving logarithms, and compare or estimate very large numbers that a calculator cannot show in full.

What this topic covers

  • The laws of logarithms
  • Change of base
  • Solving logarithmic and exponential equations
  • Comparing and estimating large powers with logarithms

Two sample questions

Sample 1

If a equals 3 to the power 239, b equals 7 to the power 139 and c equals 2 to the power 385, which of the following is true?

  1. Ab less than a less than c
  2. Bc less than a less than b
  3. Ca less than c less than b
  4. Db less than c less than a
Show the answer and solution

Answer: C

Worked solution

log⁡a\displaystyle \log a=239log⁡3,\displaystyle {}= 239 \log 3, \quadlog⁡b\displaystyle \log b=139log⁡7,\displaystyle {}= 139 \log 7, \quadlog⁡c\displaystyle \log c=385log⁡2\displaystyle {}= 385 \log 2log⁡a\displaystyle \log a≈114.0320,\displaystyle {}\approx 114.0320, \quadlog⁡b\displaystyle \log b≈117.4686,\displaystyle {}\approx 117.4686, \quadlog⁡c\displaystyle \log c≈115.8965\displaystyle {}\approx 115.8965Since log⁡x\displaystyle \log x increases with x,  a<c<b\displaystyle x, \; a < c < b

Sample 2

If the graph of log sub 10 y against log sub 10 x is a straight line with slope 3 and intercept negative 7 on the vertical axis, then y equals

  1. A10 to the power 7 x cubed .
  2. Bnegative 7x cubed .
  3. C10 to the power negative 7 x to the power negative 3 .
  4. D10 to the power negative 7 x cubed .
Show the answer and solution

Answer: D

Worked solution

log⁡10y=3log⁡10x−7\displaystyle \log_{10} y = 3\log_{10} x - 7log⁡10y\displaystyle \log_{10} y=log⁡10x3\displaystyle {}= \log_{10} x^{3}+log⁡1010−7\displaystyle {}+ \log_{10} 10^{-7}=log⁡10(10−7x3)\displaystyle {}= \log_{10}\left(10^{-7}x^{3}\right)y=10−7x3\displaystyle y = 10^{-7}x^{3}

Try a whole paper

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Original material by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.