Question

Rewrite the expression in terms of ln 2 and ln 3. ln(36)

Answer 1

The expression ln(36) can be rewritten as 2 * (ln 2 + ln 3) by using the properties of logarithms to separate the factors and bringing the exponent in front of the logarithm.

Step-by-step explanation:

To rewrite the expression ln(36) in terms of ln 2 and ln 3, we can use the property of logarithms that allows us to express ln(a*b) as ln(a) + ln(b). Since 36 can be written as 62 or (2*3)2, we can use the property of logarithms that lets us bring the exponent out in front of the ln function.

Therefore, we have:

ln(36) = ln((2*3)2)

ln(36) = 2 * ln(2*3)

ln(36) = 2 * (ln(2) + ln(3))

This is because ln(ab) = ln(a) + ln(b) and ln(ab) = b * ln(a). So, the expression ln(36) can be rewritten as 2 * (ln 2 + ln 3).

Answer 2

ln(36) can be expressed as 2 * ln(2) + 2 * ln(3).

To express ln(36) in terms of ln(2) and ln(3), you can use logarithmic properties. Remember that ln(a * b) = ln(a) + ln(b). Here's the solution:

ln(36) = ln(2² * 3²)

Using the property mentioned above:

= ln(2²) + ln(3²)

Now, applying the power rule for logarithms (n * ln(x) = ln(xⁿ)):

= 2 * ln(2) + 2 * ln(3)

So, ln(36) can be expressed as 2 * ln(2) + 2 * ln(3).