# Thread: cos and natural log

1. ## cos and natural log

Find the smallest positive number 't' such that:
10^(cos t) = 4

Round answer to three decimal places.

Thanks! =)

2. ## Re: cos and natural log

Originally Posted by jonathanraxa
Find the smallest positive number 't' such that:
10^(cos t) = 4
Round answer to three decimal places.
$\displaystyle t=\arccos\left(\frac{\ln(4)}{\ln(10)}\right)$

3. ## Re: cos and natural log

teHello, jonathanraxa!

$\displaystyle \text{Find the smallest positive number }t\text{ such that: }\,10^{\cos t} \,=\; 4$
$\displaystyle \text{Round answer to three decimal places.}$

We have: .$\displaystyle 10^{\cos t} \:=\:4$

$\displaystyle \text{Take logs, base 10: }\;\log_{10}\!\left(10^{\cos t}\right) \;=\;\log_{10}\!4$

. . . . . . . . . . . . . $\displaystyle \cos t\cdot\underbrace{\log_{10}10}_{\text{This is 1}} \;=\;\log4$

,. . . . . . . . . . . . . . . . . . $\displaystyle \cos t \;=\;\log4$

. . . . . . . . . . . . . . . . . . . . . $\displaystyle t \;=\;\cos^{-1}\!\left(\log4\right)$

.

4. ## Re: cos and natural log

Originally Posted by Soroban
teHello, jonathanraxa!

. . . . . . . . . . . . . . . . . . . . . $\displaystyle t \;=\;\cos^{-1}\!\left(\log4\right)$.
So as not to confuse the student it should be pointed out:
$\displaystyle \log_{10}(4)=\frac{\ln(4)}{\ln(10)}$.