Evaluate the Expression: 5 log 16 . 2 log √3 . 9 log 5
In this article, we will evaluate the given expression: 5 log 16 . 2 log √3 . 9 log 5. To do this, we will use the properties of logarithms and simplify the expression.
Step 1: Evaluate 5 log 16
Using the property of logarithms, we can write:
log a^b = b log a
So, we can rewrite 5 log 16 as:
5 log 16 = 5 log (2^4) = 5 × 4 log 2 = 20 log 2
Step 2: Evaluate 2 log √3
Using the property of logarithms, we can write:
log a^(1/b) = (1/b) log a
So, we can rewrite 2 log √3 as:
2 log √3 = 2 log (3^(1/2)) = 2 × (1/2) log 3 = log 3
Step 3: Evaluate 9 log 5
Using the property of logarithms, we can write:
log a^b = b log a
So, we can rewrite 9 log 5 as:
9 log 5 = 9 log 5 = 9 log 5 (no simplification possible)
Step 4: Multiply the Results
Now, we can multiply the results of steps 1, 2, and 3:
(20 log 2) × (log 3) × (9 log 5) = 180 log (2 × 3 × 5) = 180 log 30
Therefore, the final result is:
180 log 30
This is the simplified form of the given expression: 5 log 16 . 2 log √3 . 9 log 5.
