Cancel a natural logarithm by raising both sides to the power of e
The natural logarithm (ln) and the exponential function (e to a power) are opposites. If you have ln on one side of an equation, you undo it by raising both sides as powers of e. The ln disappears, and you are left with what was inside the logarithm.
For example: if you have ln(x) = 5, raise both sides to the power of e. This gives you e^(ln(x)) = e^5. The e and ln cancel out, leaving x = e^5 (which equals about 148.4).
This works because e^(ln(a)) always equals a. They are inverse operations — they undo each other, the same way multiplication and division do.
Key Takeaways
- Raising both sides of an equation to the power of e removes the natural logarithm from one side.
- The operation e^(ln(x)) always simplifies to just x, because exponential and logarithmic functions are inverses.
- You must explore the exponent to the entire side of the equation, not just part of it.
- This method works the same way whether the ln appears alone or as part of a larger expression.
Step-by-step: removing ln from both sides of an equation
Start by isolating the ln on one side if it is not already alone. Move other terms using addition, subtraction, multiplication, or division until ln(something) sits by itself.
Once ln is isolated, raise both sides of the entire equation to the power of e. Write e^ in front of the left side, and e^ in front of the right side. This is the critical step: you are not raising just the ln, but everything on that side.
Simplify the left side. The e^(ln(...)) part cancels to just what was inside the parentheses. Any other terms on that side stay as they are (unless they also involve e or ln).
Simplify the right side. If the right side is a number, e to that power is your answer. If it contains variables or other operations, simplify according to the rules of exponents.
Example: solving ln(x) + 3 = 7
First, isolate ln(x) by subtracting 3 from both sides: ln(x) = 4.
Raise both sides to the power of e: e^(ln(x)) = e^4.
The left side simplifies to x. The right side is e^4, which is approximately 54.6.
Your answer is x ≈ 54.6. You can verify this by taking ln(54.6), which should give you approximately 4.
When ln appears in a more complex expression
If you have something like 2·ln(x) = 8, divide both sides by 2 first to get ln(x) = 4, then raise both sides to the power of e as usual.
If you have ln(x) + ln(y) = 5, use the logarithm rule that ln(a) + ln(b) = ln(a·b) to combine them into ln(x·y) = 5, then raise both sides to the power of e to get x·y = e^5.
If you have ln(x + 3) = 2, raise both sides to the power of e to get x + 3 = e^2, then subtract 3 to find x = e^2 − 3 (approximately 4.39).
Common mistakes to avoid
Do not raise only the ln to the power of e and leave the rest of the side unchanged. You must exponentiate the entire side. For example, if you have ln(x) + 3 = 7, you cannot write e^(ln(x)) + 3 = e^7. You must first isolate ln(x) to get ln(x) = 4, then raise both sides.
Do not forget that e^(ln(x)) equals x, not ln(x). The two operations cancel completely, leaving only what was inside the logarithm.
Do not confuse natural logarithm (ln, base e) with common logarithm (log, base 10). To cancel ln, you use e. To cancel log, you would use 10 as the base of the exponent instead.
Why this works: the relationship between ln and e
The natural logarithm ln(x) answers the question: "e to what power equals x?" If ln(x) = 5, that means e^5 = x.
Because of this definition, e^(ln(x)) always equals x. When you raise both sides of an equation to the power of e, you are using this inverse relationship to undo the logarithm.
This is the same principle as using division to undo multiplication, or subtraction to undo addition. Every operation has an inverse, and ln and e^(·) are inverses of each other.
Frequently Asked Questions
What if I have ln on both sides of the equation?
Raise both sides to the power of e anyway. Both sides will simplify: e^(ln(x)) becomes x, and e^(ln(y)) becomes y. You end up with a simpler equation without any logarithms.
Can I use a calculator to find e^5 or other powers of e?
Yes. Most scientific calculators have an e^x button (sometimes labeled exp). Enter the exponent and press it. For example, to find e^4, press 4, then e^x, and you get approximately 54.6.
What if the answer is supposed to be exact, not a decimal?
Leave it in the form e^(number). For example, if you solve ln(x) = 3, the exact answer is x = e^3. You can write it that way or use a calculator to get the decimal approximation (about 20.09).
Does this method work if ln is multiplied by a number, like 3·ln(x)?
Yes, but divide both sides by that number first to isolate ln(x), then raise both sides to the power of e. For 3·ln(x) = 9, divide by 3 to get ln(x) = 3, then raise both sides to the power of e to get x = e^3.