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jmc

algebra junior

Problem

If

a = lo g 9

and

b = lo g 16,

compute

4^{a / b} + 3^{b / a} .

Solution

First, we have that

\frac{a}{b} = \frac{lo g 9}{lo g 16} = \frac{lo g 3 ^{2}}{lo g 4 ^{2}} = \frac{2 lo g 3}{2 lo g 4} = \frac{lo g 3}{lo g 4} .

Let

x = 4^{a / b} .

Then

lo g x = lo g 4^{a / b} = \frac{a}{b} lo g 4 = \frac{lo g 3}{lo g 4} \cdot lo g 4 = lo g 3,

so

x = 3.

Let

y = 3^{b / a} .

Then

lo g y = lo g 3^{b / a} = \frac{b}{a} lo g 3 = \frac{lo g 4}{lo g 3} \cdot lo g 3 = lo g 4,

so

y = 4.

Therefore,

x + y = 7 .

Final answer

7