Questions: If log4(-2x+8)=3, then what is the value of x?

Transcript text: If $\log _{4}(-2 x+8)=3$, then what is the value of $x$ ?

Solution

Solution Steps

To solve the equation $\log_{4}(-2x + 8) = 3$, we need to convert the logarithmic equation into an exponential form. This will allow us to solve for $x$. The equation $\log_{b}(a) = c$ can be rewritten as $b^c = a$. Applying this to our problem, we get $4^3 = -2x + 8$. We can then solve for $x$ by isolating it on one side of the equation.

Step 1: Convert the Logarithmic Equation to Exponential Form

Given the equation $\log_{4}(-2x + 8) = 3$, we convert it to exponential form. The general form $\log_{b}(a) = c$ can be rewritten as $b^c = a$. Therefore, we have: \[ 4^3 = -2x + 8 \]

Step 2: Simplify the Exponential Expression

Calculate $4^3$: \[ 4^3 = 64 \] Thus, the equation becomes: \[ 64 = -2x + 8 \]

Step 3: Solve for $x$

Rearrange the equation to solve for $x$: \[ 64 = -2x + 8 \] Subtract 8 from both sides: \[ 64 - 8 = -2x \] \[ 56 = -2x \] Divide both sides by $-2$: \[ x = \frac{56}{-2} \] \[ x = -28 \]

Final Answer

The value of $x$ is $\boxed{-28}$.

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