Suppose that In 2=a and In 11=b. Use properties of logarithms to write the logarithm in terms of a and b. In 3 square root 22.

Question

Suppose that In 2=a and In 11=b. Use properties of logarithms to write the logarithm in terms of a and b. In 3 square root 22.

Answer 1

a = ln2 ---> 2 = e^a

b = ln11 ---> 11 = e^b

ln(3√22)
= ln3 + (1/2)ln22
= ln3 + (1/2)(ln11 + ln2)
= ln3 + (b+a)/2

are you sure you typed it correctly?
I suspect we were asked for ln (2√22)

Answer 2

To write the logarithm in terms of a and b, we can use the properties of logarithms.

First, let's rewrite the given equations using the logarithmic notation:

ln(2) = a
ln(11) = b

Now, let's use the properties of logarithms to write the logarithm of √(22) in terms of a and b.

We start by using the property:

ln(ab) = ln(a) + ln(b)

ln(22) can be written as ln(2 * 11), so we can rewrite the expression as:

ln(√(22)) = ln(√(2 * 11))

Using the property we mentioned earlier, we can separate the two logarithms:

ln(√(22)) = ln(√(2)) + ln(√(11))

Since ln(2) = a and ln(11) = b, we can substitute these values:

ln(√(22)) = ln(√(2)) + ln(√(11))
= ln(2^(1/2)) + ln(11^(1/2))
= (1/2) * ln(2) + (1/2) * ln(11)
= (1/2) * a + (1/2) * b

Therefore, the logarithm of √(22) in terms of a and b is (1/2) * a + (1/2) * b.

Answer 3

To write the logarithm in terms of a and b, we can use the properties of logarithms, specifically the power and product properties.

Let's start by writing the given logarithms in exponential form using the properties of logarithms you provided:

In 2 = a can be rewritten as 2 = e^a, where e represents the constant approximating 2.71828.
In 11 = b can be rewritten as 11 = e^b.

Now, let's work on expressing the logarithm in terms of a and b for "In 3 square root 22". We will use the power and product properties of logarithms.

First, let's express the expression "3 square root 22" as an exponent. The square root of a number can be expressed as raising that number to the power of 1/2.

So, we have:
3 square root 22 = 3 * (22^(1/2))

Next, we can use the power property of logarithms, which states that log base a of (x^y) = y * log base a of x.

Applying this property to our expression, we get:
In (3 square root 22) = In (3 * (22^(1/2))) = (1/2) * In (3*22)

Now we can use the product property of logarithms, which states that log base a of (x * y) = log base a of x + log base a of y.

Applying this property to our expression again, we get:
In (3 square root 22) = (1/2) * (In 3 + In 22)

Here, we have successfully expressed the logarithm In (3 square root 22) in terms of a and b using the properties of logarithms.