The maximum value of `(log)_(20)(3sin x-4cos x+15)-` a.`\ 1` b. `2` c. `3` d. `4`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the maximum value of the expression \( \log_{20}(3\sin x - 4\cos x + 15) \), we can follow these steps:

### Step 1: Identify the expression
We start with the expression inside the logarithm:
\[ y = 3\sin x - 4\cos x + 15 \]

### Step 2: Find the maximum and minimum value of \( 3\sin x - 4\cos x \)
Using the formula for the maximum value of \( a\sin x + b\cos x \), we know that:
\[ \text{Maximum value} = \sqrt{a^2 + b^2} \]
where \( a = 3 \) and \( b = -4 \).

Calculating:
\[ \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Thus, the maximum value of \( 3\sin x - 4\cos x \) is \( 5 \) and the minimum value is \( -5 \).

### Step 3: Add 15 to the expression
Now, we add \( 15 \) to the expression:
\[ y = 3\sin x - 4\cos x + 15 \]
The maximum value of \( y \) is:
\[ 5 + 15 = 20 \]
And the minimum value of \( y \) is:
\[ -5 + 15 = 10 \]

### Step 4: Apply the logarithm
Now, we take the logarithm of \( y \):
\[ \log_{20}(y) \]
Since \( y \) varies from \( 10 \) to \( 20 \), we can find the range of \( \log_{20}(y) \).

### Step 5: Find the maximum value of \( \log_{20}(y) \)
The maximum value occurs when \( y \) is at its maximum:
\[ \log_{20}(20) \]
Using the property of logarithms:
\[ \log_{20}(20) = 1 \]

### Conclusion
Thus, the maximum value of \( \log_{20}(3\sin x - 4\cos x + 15) \) is:
\[ \boxed{1} \]