#6**+10 **

There is a way to get an exact answer, but it's not obvious, at first......

Note that we can write 81 as 3^{4} and we can write 256 as 4^{4}

And using a log property, we can write

log_{12} 81 + log_{12} 256 .....as......

log_{12} 3^{4} + log _{12} 4^{4} = log_{12}(3^{4} * 4^{4}) = log_{12}(3*4)^{4} = log_{12} (12)^{4} = 4log_{12}(12) = 4*1 = 4

And that's it.......

CPhill Jul 10, 2014

#1**+10 **

log _{12} 81 + log _{12} 256

$$\frac{log81}{log12}+\frac{log256}{log12}\\\\

=\frac{log81+log256}{log12}\\\\$$

$${\frac{\left({log}_{10}\left({\mathtt{81}}\right){\mathtt{\,\small\textbf+\,}}{log}_{10}\left({\mathtt{256}}\right)\right)}{{log}_{10}\left({\mathtt{12}}\right)}} = {\mathtt{3.999\: \!999\: \!999\: \!999\: \!999\: \!7}}$$

I suspect that the exact answer is 4. So there is probably an exact way to do this.

-----------------------

I'll think about it.

Melody Jul 10, 2014

#4**0 **

I meant that I had the answer but I wanted to see how it was supposed to be worked out.

sally1 Jul 10, 2014

#5**0 **

Yes , that is what I thought Sally.

That is good - It is nice to know that my answers have been checked - we can all make mistakes.

Melody Jul 10, 2014

#6**+10 **

Best Answer

There is a way to get an exact answer, but it's not obvious, at first......

Note that we can write 81 as 3^{4} and we can write 256 as 4^{4}

And using a log property, we can write

log_{12} 81 + log_{12} 256 .....as......

log_{12} 3^{4} + log _{12} 4^{4} = log_{12}(3^{4} * 4^{4}) = log_{12}(3*4)^{4} = log_{12} (12)^{4} = 4log_{12}(12) = 4*1 = 4

And that's it.......

CPhill Jul 10, 2014