Lesson 3: The Natural Base

New Concept

Meet the natural base, $e$, a special number for modeling real-world growth. We'll explore the natural exponential function, $f(x) = e^x$, and its inverse, the natural logarithm, $\ln x$, to solve equations and analyze continuous change.

What’s next

Now, let's put this concept into practice. You'll work through interactive examples of solving equations with $e$ and $\ln x$, and then tackle challenge problems.

Property

The number $e$ is an irrational number, where $e \approx 2.718281845$. It is often called the natural base. The natural exponential function is the function $f(x) = e^x$. Because $e$ is a number between 2 and 3, the graph of $f(x) = e^x$ lies between the graphs of $y = 2^x$ and $y = 3^x$.

Examples

• To evaluate $e^4$ using a calculator, you would find it is approximately $54.598$.
• The graph of $g(x) = e^x - 3$ is the graph of the parent function $y = e^x$ shifted 3 units down.
• The graph of $h(x) = e^{x-1}$ is the graph of the parent function $y = e^x$ shifted 1 unit to the right.

Explanation

Think of $e$ as a special number like $\pi$, but for growth. The function $f(x) = e^x$ models continuous growth, which appears everywhere in nature and finance. It's the gold standard for describing things that grow constantly.

Property

The base $e$ logarithm of a number $x$, or $\log_e x$, is called the natural logarithm of $x$ and is denoted by $\ln x$. The natural logarithm is the logarithm base $e$.

The functions $f(x) = e^x$ and $g(x) = \ln x$ are inverse functions. The following conversion formula links them:

Examples

• The logarithmic statement $\ln 15 \approx 2.708$ is equivalent to the exponential statement $e^{2.708} \approx 15$.
• The exponential statement $e^3 \approx 20.086$ is equivalent to the logarithmic statement $\ln 20.086 \approx 3$.
• We can solve $\ln x = 1$ by converting it to $e^1 = x$, so $x=e$. Also, $\ln 1 = 0$ because $e^0 = 1$.

Explanation

The natural logarithm, or $\ln x$, is the inverse of $e^x$. It answers the question: To what power must we raise $e$ to get the number $x$? This makes it essential for solving equations where the variable is in the exponent of $e$.

Property

If $x, y > 0$, then

1. Product Rule: $\ln(xy) = \ln x + \ln y$
2. Quotient Rule: $\ln \frac{x}{y} = \ln x - \ln y$
3. Power Rule: $\ln x^n = n \ln x$

Because $y = e^x$ and $y = \ln x$ are inverse functions, the following properties are also true:

Examples

• To expand $\ln(5z)$, you use the product rule: $\ln 5 + \ln z$.
• To simplify $e^{\ln(2k)}$, the inverse property applies, leaving just $2k$.
• You can simplify $\ln e^{x-5}$ using the inverse property, which gives $x-5$.

Explanation

Natural logarithms follow the same three basic rules (product, quotient, power) as any other logarithm. Additionally, their special inverse relationship with the exponential function $e^x$ means they 'undo' each other, which is a key for simplifying expressions.

Property

To solve exponential equations with base $e$, we use the natural logarithm. For an equation like $e^x = C$, we convert it to logarithmic form: $x = \ln C$. For a logarithmic equation like $\ln x = C$, we convert it to exponential form: $x = e^C$. In more complex equations, first isolate the power or the logarithm before converting.

Examples

• To solve $e^x = 40$, take the natural log of both sides: $x = \ln 40 \approx 3.6889$.
• To solve $\ln x = 2.5$, rewrite it in exponential form: $x = e^{2.5} \approx 12.1825$.
• To solve $50e^{0.1t} = 150$, first divide by 50 to get $e^{0.1t} = 3$. Then, take the natural log: $0.1t = \ln 3$, so $t = \frac{\ln 3}{0.1} \approx 10.986$.

Explanation

To solve for a variable in an exponent with base $e$, take the natural log of both sides to bring the exponent down. To solve for a variable inside a natural log, make both sides of the equation an exponent with base $e$.

Property

The function

Examples

• A city's population starts at 50,000 and grows continuously at 2% per year. The model is $P(t) = 50000e^{0.02t}$.
• A radioactive substance decays from an initial mass of 40 grams according to the formula $N(t) = 40e^{-0.05t}$. The negative value of $k$ indicates decay.
• The function $N(t) = 500(0.9)^t$ can be rewritten in base $e$. Since $k = \ln(0.9) \approx -0.1054$, the function is $N(t) \approx 500e^{-0.1054t}$.

Explanation

This formula models phenomena with continuous growth or decay, like populations or radioactive substances. The initial amount is $P_0$, and the sign of the constant $k$ determines whether the quantity is growing (positive $k$) or shrinking (negative $k$).

Property

When interest is compounded continuously, the amount $A(t)$ in an account after $t$ years is given by the function

Examples

• If you invest 2000 dollars at 4% interest compounded continuously, the value of your account after $t$ years is $A(t) = 2000e^{0.04t}$.
• To find how long it takes for a 1000 dollar investment to double to 2000 dollars at 6% interest compounded continuously, you solve the equation $2000 = 1000e^{0.06t}$.
• The value of a 5000 dollar investment after 8 years at 3.5% interest compounded continuously is $A(8) = 5000e^{0.035(8)} \approx 6615.65$ dollars.

Explanation

Continuous compounding is the theoretical limit of earning interest. It's as if interest is being calculated and added to your balance at every single moment. This formula gives the maximum amount of money an investment can earn at a given rate.