Solve for x log base 3 of x^2-2x-26=2

Problem

(log_3)(x2−2*x−26)=2

Solution

1. Rewrite the logarithmic equation in its equivalent exponential form using the definition (log_b)(y)=a⇔ba=y

x2−2*x−26=3

2. Evaluate the power on the right side of the equation.

x2−2*x−26=9

3. Set the quadratic equation to zero by subtracting 9 from both sides.

x2−2*x−35=0

4. Factor the quadratic expression by finding two numbers that multiply to −35 and add to −2

(x−7)*(x+5)=0

5. Apply the zero product property to find the potential solutions for x

x−7=0⇒x=7

x+5=0⇒x=−5

6. Verify the solutions by checking if the argument of the logarithm x2−2*x−26 is positive for both values.

For *x=7:49−14−26=9>0

For *x=−5:25+10−26=9>0

Final Answer

(log_3)(x2−2*x−26)=2⇒x=7,−5

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