ln(1/25) = ln(5^-2) = -2 ln(5)
e^(3ln2) = e^(ln8) = 8
1) ln(1/25)
2) e^3ln2 - ln5
e^(3ln2) = e^(ln8) = 8
by definition,
e^lnx = x
ln(e^x) = x
ln and e^ are inverse functions
1) ln(1/25):
We know that ln(x) represents the natural logarithm of x. In this case, we need to find ln(1/25).
Using the property of logarithms that ln(a/b) = ln(a) - ln(b), we can rewrite ln(1/25) as ln(1) - ln(25).
Since ln(1) = 0 (remember that the natural logarithm of 1 is always 0), we can simplify the expression to:
ln(1) - ln(25) = 0 - ln(25) = -ln(25).
Now let's substitute the given values:
ln(25) = ln(5^2) = 2ln(5) = 2 * 1.6 = 3.2
So ln(1/25) = -ln(25) = -3.2.
2) e^3ln2 - ln5:
We can simplify this expression using the properties of exponents and logarithms.
Using the property that e^ln(x) = x, we can simplify e^3ln2 as follows:
e^3ln2 = e^(ln2 * 3) = e^(3 * ln2).
Now, substituting the given value ln2 = 0.7 into the expression, we get:
e^(3 * ln2) = e^(3 * 0.7) = e^(2.1).
To find the value of e^(2.1), we can use the fact that e is approximately 2.71828. So:
e^(2.1) ≈ 2.71828^(2.1).
Calculating this value, we find that e^(2.1) ≈ 8.16673.
Finally, let's substitute the given value ln5 = 1.6 into the expression e^3ln2 - ln5:
e^3ln2 - ln5 = 8.16673 - 1.6 = 6.56673.
Therefore, e^3ln2 - ln5 is approximately 6.56673.