4.2. Natural Logarithms

Today, we're focusing on natural logarithms. Natural logarithms use the base e, which is approximately 2.718. Can anyone tell me what a logarithm is in general?

Isn't a logarithm about finding the exponent to which a base must be raised to produce a given number?

Exactly! So, when we say ln(x), we're finding the exponent for base e that results in x. Can anyone think of why base e is particularly important in mathematics?

I think it's because e appears in growth and decay problems, like in finance and biology.

Right! Good point. The continuous growth model is essential across various fields. So, remember: Natural logs simplify a lot of mathematical operations. Let's explore more!

Now let’s look at evaluating natural logarithms. For example, what is ln(e)?

That would be 1 because e raised to the power of 1 is e.

That's correct! ln(e) = 1. But what about ln(1)?

Isn't that 0? Because e raised to the power of 0 gives us 1!

Exactly! Remember, ln(1) = 0, and ln(e^k) = k. It simplifies our calculations immensely. Now, let's practice some evaluations!

Natural logarithms aren't just theoretical. They have real-world applications. For instance, they are used in calculating continuously compounded interest. Can anyone explain how this works?

I remember that the formula for continuously compounded interest is A = Pe^(rt).

Great! So, if we're given the total amount A and need to find r or t, how would natural logarithms help us?

We can take the natural log of both sides to solve for r or t!

Exactly! This highlights the practicality of natural logs in solving equations involving exponential growth. Let’s delve deeper into such examples.