Solve 2^x = 5
Exponential equation, worked out line by line the way a teacher would write it.
Answer
| Solution | x = ln(5)/ln(2) |
Step-by-step solution
5 steps-
1 Given2^{x} = 5
Solve for x.
-
2 Take the natural logarithm of both sides
Equal numbers have equal logarithms, so taking the natural logarithm of both sides keeps the equation true. It is the standard way to get a variable out of an exponent.
\ln{\left(2^{x} \right)} = \ln{\left(5 \right)} -
3 Use ln(bᵍ) = g · ln b to bring the exponent down
The power rule of logarithms, ln(bᵍ) = g·ln b, brings the exponent down in front, where it can be solved for like any other unknown. ln b is just a number.
x \ln{\left(2 \right)} = \ln{\left(5 \right)} -
4 Divide both sides by ln(2)
Dividing both sides by ln(2) undoes the multiplication by ln(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{\ln{\left(5 \right)}}{\ln{\left(2 \right)}} -
5 Check\begin{aligned}x = \frac{\ln{\left(5 \right)}}{\ln{\left(2 \right)}}:\quad 5 = 5\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| ln(5)/ln(2) | 5 | 5 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.