Statement 1: 5^(2log_{5}3) = 8 is False
Statement 2: 4 * ln(√e) = 2 is True
Statement 3: e^(4-ln(e^4)) = 1 is True
Statement 1: 5^(2log_{5}3) = 8
False. This statement is incorrect. The value of 5^(2log_{5}3) is 3, not 8.
Here's why:
The expression 2log_{5}3 means "3 raised to the power of 2, and then take the logarithm of that result with base 5."
So, 2log_{5}3 is equal to log_{5}(3^2).
3^2 = 9.
Taking the logarithm of 9 with base 5 (log_{5}9) is approximately 1.74.
Therefore, 5^(2log_{5}3) = 5^(1.74) which is approximately equal to 3, not 8.
Statement 2: 4 * ln(√e) = 2
True. This statement is correct.
Here's why:
ln(√e) is the natural logarithm of the square root of e.
The square root of e is approximately 1.65.
The natural logarithm of 1.65 is approximately 0.5.
Therefore, 4 * ln(√e) = 4 * 0.5 = 2.
Statement 3: e^(4-ln(e^4)) = 1
True. This statement is correct.
Here's why:
The expression ln(e^4) means "the natural logarithm of e raised to the power of 4."
e^4 is equal to e * e * e * e, which is approximately 81.
The natural logarithm of 81 is approximately 4.1.
Therefore, e^(4-ln(e^4)) = e^(4 - 4.1) = e^(-0.1).
Any number raised to the power of 0 is equal to 1, so e^(-0.1) = 1.