How would you solve different logs?

What is log #log_5^^5^7#

1 Answer
Apr 25, 2018

7

Explanation:

  1. For this question:
    The easiest way to think about logs for most people is to think of them in terms of their exponential inverses.
    For example, #log_10 100=2# because the base #(10)# raised to the answer #(2)# equals the thing you're taking the log of #(100)#. As an equation, #log_10 100=2# because #10^2=100#.
    Similarly, #log_2 8=3# because #2^3=8#.
    Looking back at your question, let's give it an answer #x#.
    Then #log_5 5^7=x#. Putting that in exponential form, #5^7 =5^x#, makes it clear that #x=7#.

  2. For other, non-standard base logs, use the change of base formula. Divide the log of the big number (here it's #5^7#) by the log of the little number (#5#). So #(log 5^7)/(log 5) =7#. You can even use natural logs for this function! #(ln 5^7)/(ln 5) =7# too.

  3. I find it handy to keep in mind the two examples I gave earlier when I'm thinking about logarithms: #log_10 100=2# and #log_2 8=3#. Writing formulas you know at the top of your paper will help you think through new problems.

Hope this helps, and Happy mathing!