Question

How do you simplify `1/2 ln (4t^4) - ln 2`?

Answer 1

This is asking you to remember the properties of logarithms. Here are the ones you need to know:

• `\mathbf(lna^b = blna)`
• `\mathbf(clna - clnb = cln\frac(a)(b))`

So, we can start by getting that exponent out in front:

`1/2ln4t^4 - ln2`

`= 1/2ln(2t^2)^2 - ln2`

Be careful that you do the above step correctly. It would be incorrect to change `ln4t^4` to `4ln4t`, because the exponent only applied to the `t` at that time.

(If you did, you would imply that the expression was `ln(4t)^4 = ln256t^4 ne ln4t^4`.)

Now, the exponent applies to the quantity `2t^2`, so we are justified in moving the power out to the front as a coefficient!

`= cancel(1/2)*cancel(2)ln2t^2 - ln2`

`= ln2t^2 - ln2`

With the same coefficients `c = 1` in front, we can now turn this into a fraction:

`= ln\frac(cancel(2)t^2)(cancel(2))`

`= lnt^2`

`= color(blue)(2lnt)`