Exponential & Logarithmic Equations Scavenger Hunt - All Things ... - Free Printable

Let’s solve the equation step by step:

We are given:

log₅(7x – 23) = log₅(3x + 1)

Since both sides have the same logarithm base (base 5), and the logs are equal, that means their arguments must be equal. That’s a key rule of logarithms: if log_b(A) = log_b(B), then A = B — as long as A > 0 and B > 0 (because you can’t take the log of zero or a negative number).

So we set the insides equal to each other:

7x – 23 = 3x + 1

Now solve for x.

Subtract 3x from both sides:

4x – 23 = 1

Add 23 to both sides:

4x = 24

Divide both sides by 4:

x = 6

✔ Now, we MUST check if this solution is valid — because logarithms only work with positive numbers.

Plug x = 6 back into the original expressions inside the logs:

Left side: 7x – 23 = 7(6) – 23 = 42 – 23 = 19 → positive ✔
Right side: 3x + 1 = 3(6) + 1 = 18 + 1 = 19 → positive ✔

Both are positive, so x = 6 is a valid solution.

Final Answer:
6