Thread: How do I do this?

1. How do I do this?

The question tells me to use the properties of logarithms and trigonometric idntities to verify the identity. *In this case, the brackets represent absolute value.

ln [sec theta] = - ln [cos theta]

Now I am not sure how to make one side equal the other.
I am thinking:
ln [1/cos theta]
but I am not sure?

Any help is appreciated.

2. Originally Posted by iluvmathbutitshard
The question tells me to use the properties of logarithms and trigonometric idntities to verify the identity. *In this case, the brackets represent absolute value.

ln [sec theta] = - ln [cos theta]

Now I am not sure how to make one side equal the other.
I am thinking:
ln [1/cos theta]
but I am not sure?

Any help is appreciated.
You are on the right track, use the log-power law to change that fraction to a exponent.

Let $sec(\theta) = u$

$ln(u) = -ln(u^{-1})$

Use the power law of logs $log_b(a^c) = c\,log_b(a)$

$ln(u) = -ln(u^{-1}) = ln([u^{-1}]^{-1}) = ln(u)$

$\therefore ln(sec\theta ) = -ln \left(cos \theta \right)$