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

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 $x$.

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: $$-3x + 14 = -4x + 16$$

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

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