Solve e^x = 5
Exponential equation, worked out line by line the way a teacher would write it.
Answer
| Solution | x = ln(5) |
Step-by-step solution
4 steps-
1 Givene^{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(e^{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(5 \right)} -
4 Check\begin{aligned}x = \ln{\left(5 \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) | 5 | 5 | ✓ |
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