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Chapter 3: Problem 30
Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=x^{6}-5 $$
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Find all numbers \(x\) that satisfy the given equation. $$ \log _{7}(x+5)-\log _{7}(x-1)=2 $$
Sirius, the brightest star that can be seen from Earth (not counting the sun), has an apparent magnitude of -1.4 . Vega, which was the North Star about 12,000 years ago (slight changes in Earth's orbit lead to changing North Stars every several thousand years), has an apparent magnitude of \(0.03 .\) How many times brighter than Vega is Sirius?
Explain how you would use a calculator to verify that $$ 2^{13746}<13746^{1000} $$ but $$ 2^{13747}>13747^{1000} $$ and then actually use a calculator to verify both these inequalities. [The numbers involved in these inequalities have over four thousand digits. Thus some cleverness in using your calculator is required.]
Show that \(\sqrt{2+\sqrt{3}}=\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}}\).
Find the number of digits in the given number. $$ 8^{4444} $$
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