Answer :

To solve the equation [tex]\log_5(x^2 - 45) = \log_5(4x)[/tex], we can use the property of logarithms that states if [tex]\log_b(A) = \log_b(B)[/tex], then [tex]A = B[/tex] provided that [tex]A > 0[/tex] and [tex]B > 0[/tex].

Step-by-step Solution:


  1. Set the arguments equal:
    Since [tex]\log_5(x^2 - 45) = \log_5(4x)[/tex], set the insides equal:
    [tex]x^2 - 45 = 4x.[/tex]


  2. Rearrange the equation:
    Move all terms to one side of the equation to form a standard quadratic equation:
    [tex]x^2 - 4x - 45 = 0.[/tex]


  3. Factor the quadratic equation:
    To factor, look for two numbers that multiply to [tex]-45[/tex] and add to [tex]-4[/tex]. These numbers are [tex]-9[/tex] and [tex]5[/tex].
    [tex](x - 9)(x + 5) = 0.[/tex]


  4. Solve for the variable [tex]x[/tex]:
    Set each factor equal to zero:
    [tex]x - 9 = 0 \quad \text{or} \quad x + 5 = 0.[/tex]
    [tex]x = 9 \quad \text{or} \quad x = -5.[/tex]


  5. Check the validity of solutions:
    Logarithms have a restriction that their argument must be positive. Check each solution:


    • For [tex]x = 9[/tex]:
      [tex]x^2 - 45 = 81 - 45 = 36[/tex] (positive) and [tex]4x = 36[/tex] (positive). Thus, [tex]x = 9[/tex] is valid.

    • For [tex]x = -5[/tex]:
      [tex]x^2 - 45 = 25 - 45 = -20[/tex] (negative) and [tex]4x = -20[/tex] (negative). Thus, [tex]x = -5[/tex] is not valid.



  6. Conclusion:
    The solution to the equation is [tex]x = 9[/tex].