Understanding Why ln(x) and e Cancel Out

[Peter G.]

Hi,

I have a hard time understanding why ln (x) and e cancel out, when, for example, we have something like: e^ln(2x+3)

I tried an internet search but I did not get any good explanation, just statements of the rule.

Could anyone help me please?

Thanks!

 

[LCKurtz]

Hi,

I have a hard time understanding why ln (x) and e cancel out, when, for example, we have something like: e^ln(2x+3)

I tried an internet search but I did not get any good explanation, just statements of the rule.

Could anyone help me please?

Thanks!

I don't like to say they "cancel out", but nevermind that. Here's one way to look at it. Say you have an equation like$$
y = e^{\ln f(x)}$$and you are wondering why $y = f(x)$. Just take the natural log of your original equation:$$
\ln y = \ln(e^{\ln f(x)})=\ln f(x) \ln e = \ln f(x)\cdot 1$$Since $y$ and $f(x)$ have equal logs, they are equal.

 

[HallsofIvy]

How are these functions "e^x" and "ln(x)" defined in your class? There are several way do define "e^x" and several different ways to define "l(x)" but which every definition of one of those is used, typically, the other is defined as its inverse function. Do you know the concept of "inverse functions"? f and g are inverse functions if and only if f(g(x))= x and g(f(x))= x. That is, the functions "cancel" each other.

If you do not understand that, before we can explain futher, we will need to know what definitions you are working with.

 

[SteveL27]

Hi,

I have a hard time understanding why ln (x) and e cancel out, when, for example, we have something like: e^ln(2x+3)

I tried an internet search but I did not get any good explanation, just statements of the rule.

Could anyone help me please?

Thanks!

e^x and ln(x) are inverse functions to each other.

Another way to say that is that ln(x) is the power you'd have to raise e to in order to get x. But then we go ahead and raise e to that power ... so we get x.

In other words, e^ln(x) = x.

Any of that correspond to what you were shown in class?

 

[ArcanaNoir]

Do you know how other logs work? Like log₁₀? You can have a log_a where "a" is any number. Now, Log₂(8)=3 because log₂(8) means 2 to the what equals 8? Well we know it is 2³=8.

Now, ln is really log_e. What happens if we take log_e(e^x)? This means e to the what equals e^x. Well, x of course. That's why ln(e^x) equals x.(expanding on what Steve said)

 

[Peter G.]

Thanks everyone! Sorry for not providing enough information at first but, studying all your responses I managed to understand.

 

[Chestermiller]

Hi,

I have a hard time understanding why ln (x) and e cancel out, when, for example, we have something like: e^ln(2x+3)

I tried an internet search but I did not get any good explanation, just statements of the rule.

Could anyone help me please?

Thanks!

The definition of the natural log ln of a number is the power that you have to raise e to in order to get that number. Therefore, ln(2x+3) is the power you have to raise e to to get 2x + 3. But in your expression, e is actually being raised to that power.