log ( 16/15 )+5 log ( 25/24 )+3 log ( 81/80 )= log (x),x

To solve the equation log (16/15) + 5 log (25/24) + 3 log (81/80) = log (x), we can use the properties of logarithms.

Step 1: Simplify the logarithmic expressions using the properties of logarithms.

Using the property log (a/b) = log(a) - log(b), we can rewrite the equation as:

log (16) - log (15) + 5(log (25) - log (24)) + 3(log (81) - log (80)) = log (x)

Step 2: Simplify further by evaluating the logarithmic expressions.

Using the property log (a^b) = b log (a), we can simplify the equation as:

log (16) - log (15) + 5(log (5^2) - log (2^3)) + 3(log (3^4) - log (2^4)) = log (x)

log (16) - log (15) + 5(2 log (5) - 3 log (2)) + 3(4 log (3) - 4 log (2)) = log (x)

Step 3: Apply the property log (a) + log (b) = log (a * b) to combine the logarithmic terms.

Using this property, we can simplify the equation as:

log (16/15) + log (5^10/2^15) + log (3^12/2^12) = log (x)

log ((16/15) * (5^10/2^15) * (3^12/2^12)) = log (x)

Step 4: Evaluate the expression inside the logarithm.

Simplifying the expression inside the logarithm, we get:

(16/15) * (5^10/2^15) * (3^12/2^12) = x

Step 5: Calculate the value of x.

Evaluating the expression, we find the value of x.