Question

How are ln x and e^x inverses of each other?

Answer 1

The natural logarithm (ln x) and the exponential function (e^x) are inverse functions, meaning they reverse each other's operations. This is captured by the equations ln(e^x) = x and e^(ln x) = x. This inverse relationship is fundamental in calculations involving growth, decay, and logarithmic properties.

Step-by-step explanation:

In mathematics, the functions ln x and e^x are closely related as they are inverse functions. This means that they effectively 'undo' each other. This inverse relationship can be demonstrated through the equations ln(e^x) = x and e^(ln x) = x. This property is particularly useful when dealing with growth and decay problems where the base of the natural logarithm, e, appears frequently.

To understand the concept further, consider e as a constant approximately equal to 2.71828. When you take the natural logarithm (ln) of a number, you're asking 'To what power do I need to raise e to get this number?' Conversely, e raised to the power of a number corresponds to the exponential growth or decay with that as the rate or time constant, depending on the context. This interplay of ln and e is fundamental to operations involving exponential growth, decay, and the logarithmic properties of multiplication.

Answer 2

Heyy there! I am AncientEnigma29 and I will be answering your question step by step

Step 1:

`\sf \: f(x) = ln(x)`

Replace f(x) with y

`\sf \: y = ln(x)`

Step 2:

Swap x and y with each other

`\tt \: x = ln(y)`

Step 3:

Solve for y...

• To do this we need to write x=ln(y) in exponential form.
• Recognize that ln(y) is a logarithm with base e, where e has approximate value of 2.71828.

`\bf \: ln(y) = log e (y)`

Rule of log used:
`\boxed{ \rm \:c = log_(a)b \: then \: {a}^(c) = b }`

By the rule we have, x=ln(y) is equal to y=e^x

Step 4:

Replace y with f^-1 (x)

`\mathcal{ {f}^( - 1)(x) = {e}^(x) }`