help with logarithms (10th grade math problem...)?

Question

so i have this logarithm:

7 ^ [ lg (lg 5) / lg 7] = ...

i have to do it step by step, so i wanted to start with solving lg5, and i found out it's 0.6989700... and how do i transform this into a fraction?? (our teacher told us it's easier with fractions than decimals) anyway, what now?

is it good if approximate it to 0.7 and do the rest like this:

7 ^ (lg 0.7/lg 7) = 7 ^ [lg (1/10)] = 7 ^ -1 = 1/7

hope i wrote things right, since I'm not from US I'm not used with this type of notations... and to think i went at a social studies profile to escape from math & physics... :((

Answer 1

Here's how it works (and to be fair, I got confused at first as well):

log x / log y = log x (y), where log x (y) means a logarithm of base x.

Step 1: Convert to a base 7 logarithm
log (log 5) / log 7 = log 7 (log 5) , where this notates that it's base 7 logarithm of log 5. Usually, the notation would have the 7 in subscript.

Step 2: Do the exponent.
7 ^ log 7 (log 5) = log 5 (solution), because taking the exponent of a logarithm of the same base will cancel it out.

ex: 10 ^ log 100 = 10 ^ 2 = 100.

Your notation, by the way, was spot on.

Answer 2

(unique) x .sixty 4^n =.a million(unique) .sixty 4^n =.a million log.sixty 4^n =log .a million nlog.sixty 4 =log .a million n =log .a million / log .sixty 4 n =5.16 6 mark downs OR .sixty 4, .sixty 4^2, .sixty 4^3, .sixty 4^4, .sixty 4^5=.107 so .sixty 4^6 is decrease than .a million