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Solve the logarithmic equation (x−4) = 3 + 7.

Mathematics

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Answer

x=25x = 25

Step-by-step solution

4 steps

  1. Apply the product rule of logarithms to the right side of the equation.

    The product rule states that log⁡M+log⁡N=log⁡(M×N)\log M + \log N = \log (M \times N).

    log⁡(x−4)=log⁡(3×7)\log(x-4) = \log (3 \times 7) log⁡(x−4)=log⁡21\log(x-4) = \log 21

  2. Since the logarithms on both sides have the same base (implied base 10), we can equate their arguments.

    x−4=21x-4 = 21

  3. Solve for xx by adding 4 to both sides of the equation.

    x=21+4x = 21 + 4 x=25x = 25

  4. Check the solution.

    The argument of a logarithm must be positive. For log⁡(x−4)\log(x-4), we need x−4>0x-4 > 0.
    If x=25x=25, then 25−4=2125-4 = 21, which is greater than 0. So the solution is valid.

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