Solve 5^(2x − 1) = 125
Exponential equation, worked out line by line the way a teacher would write it.
Answer
| Solution | x = 2 |
Step-by-step solution
7 steps-
1 Given5^{2 x - 1} = 125
Solve for x.
-
2 Isolate the exponential term5^{2 x} = 625
-
3 Write the right side as a power of 5
Powers with different bases are hard to compare. Facts like 8 = 2³ and 1/4 = 2⁻² rewrite every power with the same base 5; then the exponents can be compared, or the power replaced by a single unknown.
5^{2 x} = 5^{4} -
4 Same base on both sides, so the exponents must be equal
An exponential function with a positive base other than 1 never takes the same value twice, so bᵐ = bⁿ can only be true when m = n.
2 x = 4 -
5 Divide both sides by 2
Dividing both sides by 2 undoes the multiplication by 2, so the variable is left on its own. Both sides change in the same way, so the equation stays true (dividing by 0 is the one thing that is never allowed).
x = \frac{4}{2} -
6 Simplifyx = 2
-
7 Check\begin{aligned}x = 2:\quad 125 = 125\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| 2 | 125 | 125 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.