How do you evaluate log_169 (-13)?

Aug 29, 2016

A log expression in this form is asking the question...

"what power of 169 will give -13?"

OR" What index of 169 will make -13?"

I

Aug 29, 2016

${\log}_{169} \left(- 13\right) = \frac{1}{2} + \frac{\pi}{2 \ln \left(13\right)} i$

Explanation:

For any Real value of $x$, ${169}^{x} > 0$, so cannot be equal to $- 13$

So to find a value for ${\log}_{169} \left(- 13\right)$ we need to consider Complex logarithms.

Note that ${e}^{\pi i} = - 1$, so $\ln \left(- 1\right) = \pi i$

In general, if $x < 0$ then $\ln \left(x\right) = \ln \left\mid x \right\mid + \pi i$

Use the change of base formula to find:

${\log}_{169} \left(- 13\right)$

$= \ln \frac{- 13}{\ln} \left(169\right)$

$= \ln \frac{- 13}{\ln} \left({13}^{2}\right)$

$= \frac{\ln \left(13\right) + \pi i}{2 \ln \left(13\right)}$

$= \frac{1}{2} + \frac{\pi}{2 \ln \left(13\right)} i$

This is the principal value of the Complex logarithm.

Other values that satisfy ${169}^{z} = - 13$ are found by adding multiples of $\frac{\pi}{\ln \left(13\right)} i$

Aug 30, 2016

$\frac{1}{2} , \frac{1 + i \left(2 n + 1\right) \pi}{2 \log 13} , n = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

Explanation:

I think that I could make a compromising answer.

Use that, for $a > 0 , {a}^{2} = {\left(\pm a\right)}^{2}$ and

${\log}_{b} a = \log \frac{a}{\log} b$

Now,

${\log}_{169} \left(- 13\right) .$

$\log \frac{- 13}{\log} 169$

$= \log \frac{- 13}{\log} \left({\left(\pm 13\right)}^{2}\right)$

$= \frac{\log \frac{- 13}{\log} \left(\pm 13\right)}{2}$

$= \frac{1}{2}$, for the negative sign, and, for the positive sign,

$= \log \frac{13 {e}^{i \left(2 n + 1\right) \pi}}{2 \log 13} , n = 0 , \pm 1 , \pm 2 , \pm 3 , . .$,

using $- 1 =$cis ( odd integer multiple of pi)*

$= \frac{\log 13 + i \left(2 n + 1\right) \pi}{2 \log 13} , n = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

$= \frac{1}{2} + i \left(2 n + 1\right) \frac{\pi}{2 \log 13} , n = 0 , \pm 1 , \pm 2 , \pm 3 , . .$.

If students are not to be burdened, these questions could be

reserved for Extraordinary Talent Examinations, after doing good