# Thread: Trying to Understand Logarithm Property: Prove e^{ln(x)} = x

1. ## Trying to Understand Logarithm Property: Prove e^{ln(x)} = x

Hi,
I was presented with the following logarithm property in my textbook. I have accepted that it works and use it freely. However, I cannot prove to myself why this property works and how it is true. Would anyone please be able to give a detailed explanation and prove why this property is true?

. . . . .$e^{\ln(x)}\, =\, x$

Thanks

2. A good place to start is to refresh yourself with the definition of logarithm. The definition means that if $b^y=x$ then $log_{b}(y)=x$. Perhaps consider a few concrete examples. $2^3 = 8$ implies that $log_2(8) = 3$. Hence, what can you say about $2^{ln_2(8)}$? Similarly, $3^5 = 243$ implies that $log_3(243) = 5$. Hence, what can you say about $3^{ln_3(243)}$? What happens if you replace the 8 or the 243 in these examples with any other number? What do you notice? Finally, note that the function $ln(x)$ is defined as the logarithm base e. Now how does all of this information relate to your question?

3. Originally Posted by Onigma
I was presented with the following logarithm property in my textbook. I have accepted that it works and use it freely. However, I cannot prove to myself why this property works and how it is true. Would anyone please be able to give a detailed explanation and prove why this property is true?

. . . . .$e^{\ln(x)}\, =\, x$
There are probably loads of proofs online. Since none of them helped, and since the exercise is wanting you to understand this, let's start with The Relationship between logs, exponentials, and their bases.

. . . . .$\mbox{The equa}\mbox{tion }\, y\, =\, b^x\, \mbox{ mea}\mbox{ns the exact same}$

. . . . .$\mbox{thing as the equa}\mbox{tion }\, \log_b(y)\, =\, x.$

In this case, they've given you that the base b is the natural exponential, "e". Logs are exponents. The expression "logb(y)" means "the power y that you're putting on the base b". The equation "logb(y) = x" means "the power y, when put on the base b, evaluates to x".

If you are given the expression "ln(x)", what is the base, and what is the relationship between this base and the "x"? Where does this lead?