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jmc

algebra junior

Problem

What is the value of

lo g_{10} 4 + 2 lo g_{10} 5 + 3 lo g_{10} 2 + 6 lo g_{10} 5 + lo g_{10} 8

?

Solution

We use the two identities

a lo g_{b} x = lo g_{b} x^{a}

and

lo g_{b} x + lo g_{b} y = lo g_{b} x y

. The given expression becomes

lo g_{10} 4 + 2 lo g_{10} 5 + 3 lo g_{10} 2 + 6 lo g_{10} 5 + lo g_{10} 8 = lo g_{10} 2^{2} + lo g_{10} 5^{2} + lo g_{10} 2^{3} + lo g_{10} 5^{6} + lo g_{10} 2^{3} = lo g_{10} (2^{2} \cdot 5^{2} \cdot 2^{3} \cdot 5^{6} \cdot 2^{3}) = lo g_{10} (2^{8} \cdot 5^{8}) = lo g_{10} 1 0^{8} = 8 .

Final answer

8