3) \( (0,2)^{3 x+3}=25 \)

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To solve the equation \( (0,2)^{3x+3} = 25 \), we first need to clarify the base of the exponent. It seems there might be a typographical error, as \( (0,2) \) is not a valid base for exponentiation. Assuming you meant \( 2^{3x+3} = 25 \), we can proceed with the solution. 1. Rewrite the equation: \[ 2^{3x+3} = 25 \] 2. Take the logarithm of both sides. We can use the natural logarithm (ln) or logarithm base 10 (log). Here, we will use the natural logarithm: \[ \ln(2^{3x+3}) = \ln(25) \] 3. Apply the power rule of logarithms: \[ (3x + 3) \ln(2) = \ln(25) \] 4. Solve for \( 3x + 3 \): \[ 3x + 3 = \frac{\ln(25)}{\ln(2)} \] 5. Isolate \( 3x \): \[ 3x = \frac{\ln(25)}{\ln(2)} - 3 \] 6. Finally, divide by 3 to solve for \( x \): \[ x = \frac{1}{3} \left( \frac{\ln(25)}{\ln(2)} - 3 \right) \] Now, we can compute the numerical value of \( x \): - Calculate \( \ln(25) \) and \( \ln(2) \): \[ \ln(25) \approx 3.21888, \quad \ln(2) \approx 0.69315 \] - Substitute these values into the equation: \[ x \approx \frac{1}{3} \left( \frac{3.21888}{0.69315} - 3 \right) \] - Calculate \( \frac{3.21888}{0.69315} \): \[ \frac{3.21888}{0.69315} \approx 4.645 \] - Now substitute back: \[ x \approx \frac{1}{3} (4.645 - 3) \approx \frac{1}{3} (1.645) \approx 0.5483 \] Thus, the solution to the equation \( 2^{3x+3} = 25 \) is approximately: \[ \boxed{0.5483} \]