How does one solve a logarithmic expression where the base is a fraction? In my example I am trying to solve the following:
$$ n^{\log_\frac{3}{2}(1)} \tag{1} $$
This is related to using the "master theorem" to solve recurrence relations. People usually give examples where they solve something like:
$$ {\log_\frac{1}{3}(27)} \tag{2} $$
which seems easy to understand. However, no one really gives an example of how one would go about solving the expression listed above (1).
$\endgroup$ 22 Answers
$\begingroup$You can always change bases with the formula
$$\log_a x = \frac{\log_b x}{\log_b a}.$$
So change to your favorite integer base:
$$\log_{3/2}1 = \frac{\log_2 1}{\log_2 3/2} =\frac{\log_2 1}{\log_2 3 -1}.$$
$\endgroup$ $\begingroup$First evaluate the exponent. Here, the log of $1$ to any base is $0$, so you have $n^0=1$. If you had $n^{\log_{\frac 32}\frac 94}$ the exponent would be ${\log_{\frac 32}\frac 94}=2$ so you would have $n^2$
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