1984 AIME Problems/Problem 5
Contents
Problem
Determine the value of if and .
Solution 1
Use the change of base formula to see that ; combine denominators to find that . Doing the same thing with the second equation yields that . This means that and that . If we multiply the two equations together, we get that , so taking the fourth root of that, .
Solution 2
We can simplify our expressions by changing everything to a common base and by pulling exponents out of the logarithms. The given equations then become and $\frac{\ln b}{\ln 8} + \frac{2 \ln a}{\n 4} = 7$ (Error compiling LaTeX. Unknown error_msg). Adding the equations and factoring, we get . Rearranging we see that . Again, we pull exponents out of our logarithms to get . This means that . The left-hand side can be interpreted as a base-2 logarithm, giving us .
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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