15 Evaluate log_(2)3 __ 16 Evaluate log_(2)sqrt (16) __ 17 Given 7^7x-5=7^2x solve for x __ 18 If 9^2x=243^x+3 what is the value of x __ 19 Given log_(2)4=log_(2)9x solve for x __ 20 If log2+logx=log4 , solve for x __ 21 Solve for x: log_(5)(x+1)=log_(5)2 __ 22 Convert 230^circ into exact radians __ 23 Convert 313^0 into approximate radians __ 24 State the exact value of cos((5pi )/(3)) __ 25 State the exact value of sin((5pi )/(4)) __ 26 State the exact value of sec((7pi )/(6)) __ 27 Solve sinTheta =-(1)/(2),0leqslant Theta leqslant 2pi __ 28 Solve tanTheta =-1,0leqslant Theta leqslant 2pi __ 29 Solve cosTheta =(4)/(5),0leqslant Theta leqslant 2pi __ 30 Solve sinTheta =-(1)/(sqrt (2)),0leqslant Theta leqslant 2pi __ 31 Express sinxcos3x+cosxsin3x as single trig function __ 32 If f(x)=3sin[2(x+(pi )/(6))]-1 state phase and vertical trans __

# Explanation:## Step 1: Evaluate ### This is a logarithmic expression that cannot be simplified further without a calculator. The value of is approximately 1.58496.# Answer:### # Explanation:## Step 1: Evaluate ### Rewrite the square root as an exponent: .## Step 2: Simplify the logarithm### Use the property of logarithms: \( \log_b(a^c) = c \log_b(a) \). Thus, \( \log_2(16^{\frac{1}{2}}) = \frac{1}{2} \log_2(16) \).## Step 3: Calculate \( \log_2(16) \)### Since , we have \( \log_2(16) = 4 \).## Step 4: Final calculation### Therefore, \( \frac{1}{2} \log_2(16) = \frac{1}{2} \times 4 = 2 \).# Answer:### # Explanation:## Step 1: Given , solve for ### Since the bases are the same, set the exponents equal to each other: .## Step 2: Solve the equation### Subtract from both sides: .### Add 5 to both sides: .### Divide by 5: .# Answer:### # Explanation:## Step 1: Given , solve for ### Rewrite the bases as powers of 3: and .### Thus, \( (3^2)^{2x} = (3^5)^{x+3} \).## Step 2: Simplify the exponents### \( 3^{4x} = 3^{5(x+3)} \).### Set the exponents equal: \( 4x = 5(x + 3) \).## Step 3: Solve the equation### Distribute: .### Subtract from both sides: .### Multiply by -1: .# Answer:### # Explanation:## Step 1: Given , solve for ### Since the logarithms are equal, set the arguments equal: .## Step 2: Solve the equation### Divide by 9: .# Answer:### # Explanation:## Step 1: Given , solve for ### Use the property of logarithms: \( \log a + \log b = \log(ab) \).### Thus, \( \log(2x) = \log 4 \).## Step 2: Set the arguments equal### .## Step 3: Solve the equation### Divide by 2: .# Answer:### # Explanation:## Step 1: Solve for : \( \log _{5}(x+1)=\log _{5} 2 \)### Since the logarithms are equal, set the arguments equal: .## Step 2: Solve the equation### Subtract 1: .# Answer:### # Explanation:## Step 1: Convert into exact radians### Use the conversion factor .### Thus, .## Step 2: Simplify the fraction### .# Answer:### # Explanation:## Step 1: Convert into approximate radians### Use the conversion factor .### Thus, .# Answer:### # Explanation:## Step 1: State the exact value of \( \cos \left(\frac{5 \pi}{3}\right) \)### Recognize that is in the fourth quadrant where cosine is positive.### The reference angle is .### Thus, \( \cos \left(\frac{5 \pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \).# Answer:### \( \cos \left(\frac{5 \pi}{3}\right) = \frac{1}{2} \)# Explanation:## Step 1: State the exact value of \( \sin \left(\frac{5 \pi}{4}\right) \)### Recognize that is in the third quadrant where sine is negative.### The reference angle is .### Thus, \( \sin \left(\frac{5 \pi}{4}\right) = -\sin \left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).# Answer:### \( \sin \left(\frac{5 \pi}{4}\right) = -\frac{\sqrt{2}}{2} \)# Explanation:## Step 1: State the exact value of \( \sec \left(\frac{7 \pi}{6}\right) \)### Recognize that is in the third quadrant where cosine is negative.### The reference angle is .### Thus, \( \cos \left(\frac{7 \pi}{6}\right) = -\cos \left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} \).### Therefore, \( \sec \left(\frac{7 \pi}{6}\right) = \frac{1}{\cos \left(\frac{7 \pi}{6}\right)} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \).# Answer:### \( \sec \left(\frac{7 \pi}{6}\right) = -\frac{2\sqrt{3}}{3} \)# Explanation:## Step 1: Solve ### Identify the angles where .### These angles are in the third and fourth quadrants.### The reference angle is .### Thus, and .# Answer:### # Explanation:## Step 1: Solve ### Identify the angles where .### These angles are in the second and fourth quadrants.### The reference angle is .### Thus, and .# Answer:### # Explanation:## Step 1: Solve ### Identify the angles where .### These angles are in the first and fourth quadrants.### Using the inverse cosine function: \( \theta = \cos^{-1} \left(\frac{4}{5}\right) \) and \( \theta = 2\pi - \cos^{-1} \left(\frac{4}{5}\right) \).# Answer:### \( \theta = \cos^{-1} \left(\frac{4}{5}\right), 2\pi - \cos^{-1} \left(\frac{4}{5}\right) \)# Explanation:## Step 1: Solve ### Identify the angles where .### These angles are in the third and fourth quadrants.### The reference angle is .### Thus, and .# Answer:### # Explanation:## Step 1: Express as a single trig function### Use the sum-to-product identities: \( \sin A \cos B + \cos A \sin B = \sin(A + B) \).### Thus, \( \sin x \cos 3x + \cos x \sin 3x = \sin(x + 3x) = \sin(4x) \).# Answer:### \( \sin x \cos 3 x+\cos x \sin 3 x = \sin(4x) \)# Explanation:## Step 1: If \( f(x)=3 \sin \left[2\left(x+\frac{\pi}{6}\right)\right]-1 \), state phase and vertical translation### Identify the phase shift and vertical translation from the function.### The phase shift is given by , which means a shift of .### The vertical translation is given by the constant term outside the sine function, which is -1.# Answer:### Phase shift: , Vertical translation: -1