given that ln2=0.7 and ln5=1.6, find:

1) ln(1/25)

2) e^3ln2 - ln5

ln(1/25) = ln(5^-2) = -2 ln(5)

e^(3ln2) = e^(ln8) = 8

steve can you help me out with a question

im confused on how you got 8 for the second problem. may you please explain?

3ln2 = ln 2^3 = ln8

by definition,
e^lnx = x
ln(e^x) = x

ln and e^ are inverse functions

To find the values of ln(1/25) and e^3ln2 - ln5 using the given information, we can use the properties of logarithms and exponents.

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.