solve the equation for ( 0 leq x<2 pi ).\n( 6^{cos x}=5 )\nselect the correct choice below and, if…

solve the equation for ( 0 leq x<2 pi ).\n( 6^{cos x}=5 )\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. ( x= )\n(round to four decimal places as needed. use a comma to separate answers as needed.)\nb. there is no solution.

solve the equation for ( 0 leq x<2 pi ).\n( 6^{cos x}=5 )\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. ( x= )\n(round to four decimal places as needed. use a comma to separate answers as needed.)\nb. there is no solution.

Answer

Explanation:

Step1: Take the natural logarithm of both sides

$$\ln(6^{\cos x})=\ln(5)$$ Using the property $\ln(a^b)=b\ln(a)$, we get $\cos x\ln(6)=\ln(5)$.

Step2: Solve for $\cos x$

$$\cos x=\frac{\ln(5)}{\ln(6)}$$ Calculate $\frac{\ln(5)}{\ln(6)}\approx0.8982$.

Step3: Find $x$ values in $[0, 2\pi)$

Since $\cos x = 0.8982$, then $x=\cos^{-1}(0.8982)$ or $x = 2\pi-\cos^{-1}(0.8982)$. Calculate $\cos^{-1}(0.8982)\approx0.4556$, and $2\pi - 0.4556\approx5.8276$.

Answer:

A. (x = 0.4556,5.8276)