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smc

algebra senior

Problem

If

lo g_{10} 2 = a

and

lo g_{10} 3 = b

, then

lo g_{5} 12 =

?

(A)

\frac{a + b}{a + 1}

(B)

\frac{2 a + b}{a + 1}

(C)

\frac{a + 2 b}{1 + a}

(D)

\frac{2 a + b}{1 - a}

Solution

From the change of base formula,

lo g_{5} 12 = \frac{l o g _{10} 12}{l o g _{10} 5}

. For the numerator,

2 \cdot 2 \cdot 3 = 12

, so

lo g_{10} 12 = lo g_{10} 2 + lo g_{10} 2 + lo g_{10} 3 = 2 a + b

. For the denominator, note that

lo g_{10} 10 = 1

. Thus,

lo g_{10} 5 = lo g_{10} 10 - lo g_{10} 2 = 1 - a

. Thus, the fraction is

\frac{2 a + b}{1 - a}

, so the answer is

\frac{2 a + b}{1 - a}

.

Final answer

D