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Logarithmic Functions

Logarithmic scales are used when the range of possible values is very wide, such as the intensity of an earthquake or the acidity of a liquid, in order to avoid the use of large numbers.

Compound Problems


\(f(x) = \log_{60}(x^2)\)

What is the value of \(f(3)+f(4)+f(5)\) ?

\[\log_2(\sqrt{x}) + \log_4(x-3) = \log_{16}(9x^2)\]

What value(s) of x satisfy this equation?

\[2(\log_x(3)) + \log_3(x) = 3\]

What is the sum of all real values of \(x\) that satisfy the above equation?

\(f(x) = \log_{2}(x) - \log_{2^2}(x)\) \(+ \log_{2^3}(x) - \log_{2^4}(x) \dots\) \(+ \log_{2^{99}}(x) - \log_{2^{100}}(x)\)

What is the value of \(f(2^{(2^{100})}) - (2)f(2^{(2^{99})})\) \(+ (2^2)f(2^{(2^{98})}) - (2^3)f(2^{(2^{97})}) \dots\) \(+ (2^{98})f(2^{(2^{2})}) - (2^{99})f(2^{(2^{1})})\) ?

Let \(f(x) = \log_{10}(\log_{10}(\log_{10}(x)))\)

The domain of \(f\) is \((k,\infty)\)

What is the value of \(k\)?


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