Solve for x log of x^2-3x=1

Problem

(log_)(x2−3*x)=1

Solution

1. Identify the base of the logarithm. Since no base is explicitly written, it is assumed to be the common logarithm with base 10

(log_10)(x2−3*x)=1

2. Rewrite in exponential form to remove the logarithm. The equation (log_b)(y)=a is equivalent to ba=y

x2−3*x=10

3. Set the quadratic equation to zero by subtracting 10 from both sides.

x2−3*x−10=0

4. Factor the quadratic expression by finding two numbers that multiply to −10 and add to −3 These numbers are −5 and 2

(x−5)*(x+2)=0

5. Solve for x by setting each factor equal to zero.

x−5=0⇒x=5

x+2=0⇒x=−2

6. Verify the solutions by checking if the argument of the logarithm, x2−3*x is positive for both values.

For *x=5:5−3*(5)=25−15=10>0

For *x=−2:(−2)2−3*(−2)=4+6=10>0

Final Answer

(log_)(x2−3*x)=1⇒x=5,−2

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