Solve the following logarithmic equation. Show step by step work.

Question

Solve the following logarithmic equation. Show step by step work.

log4 (𝑥 − 3) = 2 − log4 (𝑥 + 12)

Answer 1

To solve for x in the equation log₄(x-3) = 2 - log₄(x+12), we can first bring both logs to the same side by adding log₄(x+12) to the other side so it becomes:

log₄(x-3) + log₄(x+12) = 2

Since both logs have the same base (4) and are being added, we can apply the logarithmic product property which is:

log_b(mn) = log_bm + log_bn

Basically, according to the property, if two logs are being multiplied and they have the same base, we can break them down into two logs that are added together. We can apply the same rule vice versa. When we apply the property to our problem, it becomes:

log₄(x-3)(x+12) = 2

To solve this further, we can convert this to exponential form:

4² = (x-3)(x+12)

Now we just do the algebra required to solve for x by following these steps:

16 = x² + 12x - 3x - 36 Simplified by doing 4² = 16 and foiling the right side

16 = x² + 9x - 36 Simplified further by adding like terms

0 = x² + 9x - 52 Brought 16 to the other side by subtracting by 16

0 = (x - 4)(x + 13) Factored the quadratic

x - 4 = 0 x + 13 = 0 Use the zero-product rule to set each factor to 0 and solve for x

x = 4 x = -13

Lastly, we do need to plug these values back into the equation to make sure they make the equation true. We can take the rearranged equation to make it easy for ourselves to plug in the value and solve which is this one:

log₄(x-3) + log₄(x+12) = 2

First, we can plug in x = 4 and solve both sides to see if the equation is true:

log₄(4-3) + log₄(4+12) = 2 Right side is just 2 so don't need to solve but we can simplify left side

log₄(1) + log₄(16) = 2

0 + 2 = 2

2 = 2

So, x = 4 is a solution since the equation came out true (2 = 2) when we plugged it in.

Now, we can plug in x = -13 into the equation:

log₄(-13-3) + log₄(-13+12) = 2

log₄(-16) + log₄(-1) = 2

The log of a number is undefined. Therefore, x = -13 is not a solution.

The only solution of the equation that makes the equation true is x = 4.

Answer 2

log (ab) = log (a) + log (b). so

log₄(x - 3) + log₄ ( x + 12) = 2

then log₄ (x -3)(x +12) = 2

Now convert to exponential form

4² = (x -3)(x + 12)

16 = x² + 9x -36

0 = x² + 9x -52 = (x + 13)(x - 4)------>. x= -13 or 4. but check if answers work in the orig. problem

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