(2*log7 16)/(log3(√10+1)+log3(√10-1)log7 2
Fractions have implicit parenthese around the denominator and numerator.
I have a feeling that the two log3 terms both belong to the denominator, in which case the expression should have been written, according to the BEDMAS rules,
(2*log7 16)/((log3(√10+1)+log3(√10-1)log7 2 )
in which case:
(2*log724)/((log3(√10+1)(√10-1) log7 2 )
=(8*log72)/((log3(10-1) log7 2 )
=(8*log72)/((log3(9) log7 2 )
=(8*log72)/(2 log7 2 )
=4
I suspect a typo, since
log3(√10+1)+log3(√10+1) = log3(10-1) = log3(9) = 2
the log7 of powers of 2 is not amenable to simplification.
Is there a typo? At the very least, the parentheses are not balanced.
Just a lucky guess, but you were right in the first place!
To simplify the expression (2log7 16)/(log3(√10+1) + log3(√10-1)log7 2), let's break it down step by step.
Step 1: Let's start by simplifying the logarithmic terms.
Using the following logarithmic properties:
- log a^b = b*log a
- log a + log b = log(ab)
- log a - log b = log(a/b)
- log a + log b = log a*b
We can rewrite the expression as:
(2 * log7 16) / (log3(√10+1) + log3(√10-1)log7 2)
= (2 * log7 (4^2)) / (log3(√10+1) + log3(√10-1)log7 2)
= (2 * 2 * log7 4) / (log3(√10+1) + log3(√10-1)log7 2)
= (4 * log7 4) / (log3(√10+1) + log3(√10-1)log7 2)
Step 2: Simplify the logarithmic terms further.
Using the property log a^b = b*log a, we have:
- log7 4 = log7 (2^2) = 2 * log7 2
Therefore, we can rewrite the expression as:
(4 * 2 * log7 2) / (log3(√10+1) + log3(√10-1)log7 2)
= (8 * log7 2) / (log3(√10+1) + log3(√10-1)log7 2)
Step 3: Combine the logarithmic terms.
Using the property log a + log b = log(a*b), we can simplify the expression further:
(8 * log7 2) / (log3(√10+1) + log3(√10-1)log7 2)
= (8 * log7 2) / (log3(√10+1) + log3(√10-1) + log7 2)
Step 4: Simplify the expression inside the logarithmic terms.
Using the property √a = a^(1/2), we have:
- √10 + 1 = (10^(1/2)) + 1
- √10 - 1 = (10^(1/2)) - 1
Therefore, we can rewrite the expression as:
(8 * log7 2) / (log3((10^(1/2)) + 1) + log3((10^(1/2)) - 1) + log7 2)
This is the simplified form of the expression.