Logarithms are mathematical functions that help us solve exponential equations. They represent the exponent to which a base number must be raised to obtain a given value. In this problem, we are asked to simplify the expression: log49√7 log25√5-log4√2/log17.5.


To begin solving this expression, we need to simplify each term individually. Let's start with the first term: log49√7.


The expression log49√7 can be rewritten as log49(7^(1/2)). Using the property of logarithms, we know that log_a(b^c) = c * log_a(b). Applying this property, we can simplify the expression as (1/2) * log49(7).


To further simplify the expression, we need to find the base of the logarithm, which is 49. We can rewrite 49 as 7^2, since 7^2 equals 49. Therefore, log49(7) is equivalent to log(7^(1/2)) / log(7^2).


Using the property of logarithms again, we can rewrite the expression as (1/2) * (log(7) / log(7^2)). Now, we can simplify further by evaluating log(7) and log(7^2).


Using a calculator or logarithm tables, we find that log(7) is approximately 0.8451 and log(7^2) is approximately 1.2041. Plugging these values back into our expression, we have (1/2) * (0.8451 / 1.2041).


Similarly, we can simplify the second term log25√5 as log25(5^(1/2)). Following the same steps as above, we find that this term simplifies to (1/2) * (log(5) / log(5^2)).


The third term log4√2 can be simplified as log4(2^(1/2)). Applying the same steps, we find that this term simplifies to (1/2) * (log(2) / log(2^2)).


Now that we have simplified each term, we can combine them into a single expression. The expression becomes ((1/2) * (0.8451 / 1.2041)) - ((1/2) * (log(5) / log(5^2))) - ((1/2) * (log(2) / log(2^2))).


To simplify further, we can combine the fractions by finding a common denominator. The common denominator in this case is log(5^2) * log(2^2).


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