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Logarithmic Inequalities

Logarithmic inequalities are useful in analyzing situations involving repeated multiplication, such as interest and exponential decay.

Warmup

True or false: \(\ln(10) > 2\)

\[\log_{2}(x-2) + \log_{2}(x-3) < 1\]

What values of \(x\) satisfy the above statement?

How many integer values of \(x\) satisfy this inequality?

\[\ln(2x) > \ln(x^2-3)\]

What values of \(x\) satisfy the relationship

\(\log_{6}(x-3) < \log_{7}(x-3)\)

\[\log_{5}(x-5) < 2\]

Solve for x.

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