Question 635629
I'm not sure I understand what your question is.<br>
If the question is "How does one find the value of a logarithm?", then there are two main options:<ul><li>On your own, without help. When finding logarithms on your own they break down into two groups:<ul><li>Logarithms that can be found with a reasonably small amount of effort. In this category would be logs like {{{log(2, (8))}}} or {{{log(9, (81))}}} or {{{log(16, (4))}}} (1/2), etc. These logs are relatively easy because it is not too difficult to figure out what power of the base results in the argument.</li><li>Logarithms which would require a major effort to calculate. In this group would be logs like log(7). It is not easy to find out what power of 10 results in a 7. Since {{{10^0 < 7 < 10^1}}} you know that log(7) will be between 0 and 1. After that it get difficult. One way to find log's (or sin's, cos's, etc.) is to use what are called "Taylor's series". (There may be other ways but I cannot think of them at the moment.) Taylor's series use advanced Math (Calculus) to "convert" something like a log (or sin or cos, etc.) into an expression  for a decimal approximation that could be calculated by hand. If you want to know more and you know about derivatives then look up "Taylor's series".</li></ul></li><li>With help. Use a calculator or a table of logarithms. This is easy but keep in mind that the calculator will give you decimal approximations most of the time. (This is why {{{10^0.8451}}} is not <i>exactly</i> 7.) And when it does give you an exact answer it may be difficult to tell that it is not an approximation. So I recommend finding logs by hand when possible and using calculators for the rest.</li></ul>