Studdy AI Logo
Math  /  Algebra

QuestionUse the properties of logarithms to simplify the expression ln⁡(sin⁡θ)−ln⁡(sin⁡θ4)\ln (\sin \theta)-\ln \left(\frac{\sin \theta}{4}\right). ln⁡(sin⁡θ)−ln⁡(sin⁡θ4)=□\ln (\sin \theta)-\ln \left(\frac{\sin \theta}{4}\right)=\square (Type an exact answer.)

Studdy Solution

STEP 1

1. We are asked to simplify a logarithmic expression.
2. The properties of logarithms can be used to simplify the expression.

STEP 2

1. Apply the logarithmic property for subtraction.
2. Simplify the resulting expression.

STEP 3

Use the logarithmic property that states:
ln⁡(a)−ln⁡(b)=ln⁡(ab) \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)
Apply this property to the expression:
ln⁡(sin⁡θ)−ln⁡(sin⁡θ4)=ln⁡(sin⁡θsin⁡θ4) \ln (\sin \theta) - \ln \left(\frac{\sin \theta}{4}\right) = \ln \left(\frac{\sin \theta}{\frac{\sin \theta}{4}}\right)

STEP 4

Simplify the expression inside the logarithm:
sin⁡θsin⁡θ4=sin⁡θ×4sin⁡θ=4 \frac{\sin \theta}{\frac{\sin \theta}{4}} = \frac{\sin \theta \times 4}{\sin \theta} = 4
Thus, the expression simplifies to:
ln⁡(4) \ln(4)
The simplified expression is:
ln⁡(4) \boxed{\ln(4)}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord