Question

How do you solve `2^ logx = 1/4`?

Answer 1

`x = 1/100` or `x = 1/e^2` depending on whether `log` refers to `log_10` or `log_e`.

Explanation:

We will use the following:

• `log_a(a) = 1`

• `log(a^x) = xlog(a)`

• `a^(log_a(x)) = x`

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`2^log(x) = 1/4 = 2^(-2)`

`=> log_2(2^log(x)) = log_2(2^(-2))`

`=> log(x)log_2(2) = -2log_2(2)`

`=>log(x) = -2`

(Depending on the context, `log` may refer to `log_10` or `log_e`. We will show both cases.)

`=> 10^(log_10(x)) = 10^(-2)` or `e^(log_e(x)) = e^(-2)`

`=>x = 10^(-2) = 1/100` or `x = e^-2 = 1/e^2`