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Answer
\( x=1 \)
Solution
Solve the equation \( \log_{2}(-5x+9)=2 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\log_{2}{\left(-5x+9\right)}=2\)
- step1: Find the domain:
\(\log_{2}{\left(-5x+9\right)}=2,x<\frac{9}{5}\)
- step2: Convert the logarithm into exponential form:
\(-5x+9=2^{2}\)
- step3: Evaluate the power:
\(-5x+9=4\)
- step4: Move the constant to the right side:
\(-5x=4-9\)
- step5: Subtract the numbers:
\(-5x=-5\)
- step6: Change the signs:
\(5x=5\)
- step7: Divide both sides:
\(\frac{5x}{5}=\frac{5}{5}\)
- step8: Divide the numbers:
\(x=1\)
- step9: Check if the solution is in the defined range:
\(x=1,x<\frac{9}{5}\)
- step10: Find the intersection:
\(x=1\)
The solution to the equation \( \log_{2}(-5x+9)=2 \) is \( x=1 \).
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Simplify this solution The Deep Dive
To solve the equation \( \log_{2}(-5x + 9) = 2 \), we first convert the logarithmic equation into its exponential form. This gives us: \[ -5x + 9 = 2^2 \] Calculating the right side: \[ -5x + 9 = 4 \] Next, we'll isolate the variable \( x \). We start by subtracting 9 from both sides: \[ -5x = 4 - 9 \] This simplifies to: \[ -5x = -5 \] Now, divide both sides by -5: \[ x = 1 \] Now, let's check if this solution is valid by substituting \( x = 1 \) back into the original logarithmic equation: \[ \log_{2}(-5(1) + 9) = \log_{2}(4) \] Since \( \log_{2}(4) = 2 \), the solution is confirmed. Therefore, the solution is: \[ \boxed{1} \]
