Which of the following expressions is equal to \log _{10}(10 \times 9 \times 8 \times \cdots \times 2 \times 1) ?

Correct Answer is : 2+2log102+4log106+log107,

1+5 \log _{10} 2+4 \log _{10} 6,

1+4 \log _{10} 2+2 \log _{10} 6+\log _{10} 7,

2+6 \log _{10} 2+4 \log _{10} 6+\log _{10} 7,

2+6 \log _{10} 2+4 \log _{10} 6.

Oxford University MAT 2020 Solved Papers

Question : 6 of 10

Marks: +1, -0

Which of the following expressions is equal to

\log _{10}(10 × 9 × 8 × ⋯ × 2 × 1)

Solution:

Factorise the numbers from 1 to 10 ; this gives

\log _{10}(2 × 5 × 3^2 × 2^3 × 7 × 2 × 3 × 5 × 2^2 × 3 × 2)=\log _{10}(2^8 3^4 5^2 7)

Use

\log _{10}(2^2 5^2)=2

. Use

2^4 3^4=6^4

. This gives

2+\log _{10}(2^2 6^4 7)=2+2 \log _{10} 2+4 \log _{10} 6+\log _{10} 7

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