Questions: Solve for (x). [ log 5(-3 x+8)=1 ] [ x= ]

Solve for (x). [ log 5(-3 x+8)=1 ] [ x= ]
Transcript text: Solve for $x$. \[ \log _{5}(-3 x+8)=1 \] \[ x= \]
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Solution

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Solution Steps

Step 1: Understand the Logarithmic Equation

The given equation is:

\[ \log_{5}(-3x + 8) = 1 \]

This equation states that the logarithm base 5 of \(-3x + 8\) is equal to 1.

Step 2: Convert the Logarithmic Equation to Exponential Form

To solve the equation, we convert the logarithmic equation to its equivalent exponential form. The equation \(\log_{b}(a) = c\) can be rewritten as \(b^c = a\).

Applying this to our equation:

\[ 5^1 = -3x + 8 \]

This simplifies to:

\[ 5 = -3x + 8 \]

Step 3: Solve for \(x\)

Now, solve the equation for \(x\):

  1. Subtract 8 from both sides:

    \[ 5 - 8 = -3x \]

    \[ -3 = -3x \]

  2. Divide both sides by \(-3\):

    \[ x = \frac{-3}{-3} \]

    \[ x = 1 \]

Final Answer

The solution to the equation is:

\[ \boxed{x = 1} \]

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