Questions: Solve: log4(-3x+14)=log4(-4x+16) x=□ (Enter DNE if no solution exists)

Solve: log4(-3x+14)=log4(-4x+16) x=□ (Enter DNE if no solution exists)
Transcript text: Solve: \[ \begin{array}{l} \log _{4}(-3 x+14)=\log _{4}(-4 x+16) \\ x=\square \end{array} \] (Enter DNE if no solution exists)
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Solution

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Solution Steps

To solve the equation \(\log_{4}(-3x+14) = \log_{4}(-4x+16)\), we can use the property of logarithms that states if \(\log_b(A) = \log_b(B)\), then \(A = B\). This allows us to set the arguments of the logarithms equal to each other and solve the resulting linear equation for \(x\).

Step 1: Set Up the Equation

We start with the equation given by the logarithmic equality: \[ \log_{4}(-3x + 14) = \log_{4}(-4x + 16) \] Using the property of logarithms, we can equate the arguments: \[ -3x + 14 = -4x + 16 \]

Step 2: Solve for \(x\)

Rearranging the equation, we have: \[ -3x + 4x = 16 - 14 \] This simplifies to: \[ x = 2 \]

Step 3: Verify the Solution

We need to ensure that the solution \(x = 2\) is valid within the domain of the logarithmic functions. We check the arguments:

  1. For \(-3x + 14\): \[ -3(2) + 14 = -6 + 14 = 8 > 0 \]
  2. For \(-4x + 16\): \[ -4(2) + 16 = -8 + 16 = 8 > 0 \] Both arguments are positive, confirming that \(x = 2\) is valid.

Final Answer

\[ \boxed{x = 2} \]

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