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Exponential Inequalities

When will a bacteria colony reach a certain size? When will one investment outperform another? Many real-world models use exponential functions, and you'll need these tools to compare them.

Multiple Terms

         

Solve the following for \(x:\) \[\left\{\begin{matrix} 3^x > 9 \\ 3^{x^2} \leq 27^{4x}. \end{matrix}\right.\]

What is the range of \(x\) that satisfies the inequality \( \left ( \frac{1}{49} \right )^{x+4} + \left ( \frac{1}{7} \right )^{x+4} > 12 ?\)

What is the range of \(x\) that satisfies the inequality \( \left(\frac{1}{5}\right)^{3+x} < 625 < \left( \frac{1}{25} \right)^x ?\)

Given \(\log 2=0.3010\) and \(\log 3=0.4771,\) solve the inequality \[\left(2 ^{7}\right)^x<\left(3 ^x\right)^{6}.\]

Given \(n \geq 2,\) which of the following numbers is the largest: \[ \sqrt[n]{6^{n-1}}, \sqrt[n]{6^{n+1}}, \sqrt[n+1]{6^n},6 ?\]

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