How do you approximate #log_7 16# given #log_7 2=0.3562# and #log_7 3=0.5646#?

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Answer 1 · 2016-11-14T14:26:55

We have that #16 = 2^4#, or #2 xx 2 xx 2 xx 2#. Writing in logarithms, we have:

#log_7(16) = log_7(2 xx 2 xx 2 xx 2)#

Using the sum rule of logarithms that #log_an + log_am = log_a(n xx m)#, we have:

#=>log_7 2 + log_7 2 + log_7 2 + log_7 2#

#=>0.3562 + 0.3562 + 0.3562 + 0.3562#

#=> 1.4248#

Hence, #log_7 16~=1.4248#. Checking with a calculator gives #log_7 16 ~= 1.424828748#. So, our estimate was correct to 4 decimals.

Hopefully this helps!