1964 AHSME Problems/Problem 21
Contents
[hide]Problem 21
If , then equals:
Solution 1
Using natural log as a "neutral base", and applying the change of base formula to each term, we get:
You could inspect the equation here and see that is one solution. Or, you can substitute and to get a quadratic in :
The above is a quadratic with coefficients . Plug into the QF to get:
Either way, the answer is .
Solution 2
All answers are of the form , so we substitute that into the equation and try to solve for . We get:
By the definition of a logarithm, the first term on the left is asking for the exponent needed to change the number into to get to . That exponent is .
The second term is asking for a similar exponent needed to change into . That exponent is .
The equation becomes . Multiplying by gives the quadratic , which has the solution . Thus, , and the answer is .
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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