The number e and the natural logarithm

Exponential and logarithmic functions, grade 12 (5 units) Lesson 10 of 23Learning

The number e and the natural logarithm

Interest worked out more often

Three banks offer to hold 1,000 ₪ for one year:

Plenty Bank: interest of 100% a year. At the end of the year – $\displaystyle {1000\cdot2}={2000}$ ₪.
Growth Bank: interest of 50% for half a year, twice. At the end of the year – $\displaystyle {1000\cdot1.5^2}={2250}$ ₪.
Success Bank: interest of 25% for three months, four times. At the end of the year – $\displaystyle 1000\cdot1.25^4\approx2441.41$ ₪.

The more periods the year is split into, the larger the sum – but not without limit. Split into $\displaystyle n$ periods, the sum is $\displaystyle 1000\cdot\left(1+\frac{1}{n}\right)^n$ :

$\displaystyle {n}={12}$ (every month): $\displaystyle \approx2613.04$ ₪.
$\displaystyle {n}={365}$ (every day): $\displaystyle \approx2714.57$ ₪.
$\displaystyle n$ very large: the sum comes close to $\displaystyle 2718.28$ ₪.

The number that $\displaystyle \left(1+\frac{1}{n}\right)^n$ comes close to is called $\displaystyle e$ : $\displaystyle e\approx2.71828$ . It is a constant, like $\displaystyle \pi$ , and not a fraction.

\displaystyle e^x

and the natural logarithm

\displaystyle \ln x

$\displaystyle e^x$ is the exponential function whose base is $\displaystyle e$ . The logarithm to the base $\displaystyle e$ is called the natural logarithm, written $\displaystyle \ln$ :

\displaystyle \ln x=\log_{e}x

$\displaystyle e^x$ and $\displaystyle \ln x$ are inverses of each other, so their graphs are symmetric about the line $\displaystyle {y}={x}$ :

\displaystyle f(x)=e^{x}

\displaystyle g(x)=\ln x

\displaystyle y=x

$\displaystyle {e^{\ln x}}={x}$ for every $\displaystyle x>0$ , and $\displaystyle {\ln e^x}={x}$ for every $\displaystyle x$ .
$\displaystyle {\ln1}={0}$ and $\displaystyle {\ln e}={1}$ .
The laws of logarithms hold for $\displaystyle \ln$ too: $\displaystyle {\ln(xy)}={\ln x+\ln y}$ , $\displaystyle {\ln\frac{x}{y}}={\ln x-\ln y}$ , $\displaystyle {\ln x^b}={b\ln x}$ .
A calculator has an $\displaystyle \ln$ key: $\displaystyle \ln2\approx0.693$ .

Every exponential function to the base

\displaystyle e

Every positive number $\displaystyle a$ equals $\displaystyle e^{\ln a}$ , so:

\displaystyle a^{x}

\displaystyle =

\displaystyle e^{x\ln a}

$\displaystyle {2^x}={e^{x\ln2}\approx e^{0.693x}}$ .
A process of growth and decay $\displaystyle {f(t)}={f(0)\cdot q^t}$ can be written $\displaystyle {f(t)}={f(0)\cdot e^{ct}}$ , where $\displaystyle {c}={\ln q}$ .
$\displaystyle c>0$ (that is $\displaystyle q>1$ ) – growth; $\displaystyle c<0$ (that is $\displaystyle 0<q<1$ ) – decay. Interest of 5% a year: $\displaystyle {c}={\ln1.05\approx0.0488}$ .

Equations with

\displaystyle e

and

\displaystyle \ln

The examples below solve equations by the definition of $\displaystyle \ln$ and by substitution:

$\displaystyle {3e^x}={12}$ : $\displaystyle {e^x}={4}$ , so $\displaystyle {x}={\ln4}$ .
$\displaystyle {\ln(x+1)}={2}$ : $\displaystyle {x+1}={e^2}$ , so $\displaystyle {x}={e^2-1}$ . It is in the domain, because $\displaystyle {x+1}={e^2>0}$ .
$\displaystyle {e^{2x}-5e^x+6}={0}$ : we substitute $\displaystyle {t}={e^x}$ and get $\displaystyle {t^2-5t+6}={0}$ , that is $\displaystyle {t}={2}$ or $\displaystyle {t}={3}$ .
$\displaystyle {e^x}={2}$ or $\displaystyle {e^x}={3}$ , so $\displaystyle {x}={\ln2}$ or $\displaystyle {x}={\ln3}$ .

The logarithmic scale

In growth $\displaystyle {y}={A\cdot e^{ct}}$ the values of $\displaystyle A$ and $\displaystyle c$ are hard to read off the graph. Drawing $\displaystyle \ln y$ instead of $\displaystyle y$ gives a straight line:

\displaystyle \ln y=\ln A+ct

The slope of the line is $\displaystyle c$ .
The line meets the vertical axis at $\displaystyle \ln A$ .

\displaystyle y=0.5x+3

\displaystyle (0,3)

\displaystyle (4,5)

In a culture of bacteria the number of bacteria $\displaystyle y$ was measured, and $\displaystyle \ln y$ was drawn as a function of the time $\displaystyle t$ in hours. The result was a line through $\displaystyle (0,3)$ and $\displaystyle (4,5)$ . The example below finds $\displaystyle A$ and $\displaystyle c$ .

The slope: $\displaystyle {c}={\frac{5-3}{4}}={0.5}$ .
$\displaystyle {\ln A}={3}$ , so $\displaystyle {A}={e^3\approx20.09}$ .
$\displaystyle {y}={e^3\cdot e^{0.5t}}$ : at the start there were about 20 bacteria.

Exercise 1

Calculate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Solve the equations.

Exercise 3

Write $\displaystyle 5^x$ as a power of $\displaystyle e$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

The number of bacteria in a culture is $\displaystyle {y}={A\cdot e^{ct}}$ , where $\displaystyle t$ is the time in hours.
The graph of $\displaystyle \ln y$ as a function of $\displaystyle t$ holds the points $\displaystyle (0,2)$ and $\displaystyle (4,3.2)$ .

Write the equation of the line through the two points.

\displaystyle (0,2)

\displaystyle (4,3.2)

Calculate $\displaystyle c$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Calculate $\displaystyle A$ , and round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Find the doubling time of the number of bacteria: solve the equation.

Formulas: exponential and logarithmic functions, grade 12 (5 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
Change of base
$\displaystyle \log_{c}x$ $\displaystyle =$ $\displaystyle \frac{\log_{a}x}{\log_{a}c}$
$\displaystyle \log_{a}b$ $\displaystyle =$ $\displaystyle \frac{1}{\log_{b}a}$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

The exponential and the logarithmic function

Every exponential as a power of $\displaystyle e$
$\displaystyle a^{x}$ $\displaystyle =$ $\displaystyle e^{x\ln a}$
$\displaystyle a^x$ and $\displaystyle \log_a x$ – inverse functions
$\displaystyle y=a^{x}\iff x=\log_{a}y$ $\displaystyle y>0$
The number $\displaystyle e$
$\displaystyle e\approx2.718$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$
The doubling time $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle 2$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (a^{x})^{\prime}$ $\displaystyle =$ $\displaystyle a^{x}\cdot\ln a$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
$\displaystyle (\log_{a}x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x\cdot\ln a}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals, the area and accumulation

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int a^{x}\,dx=\frac{a^{x}}{\ln a}+C$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
Where $\displaystyle f(x)\neq0$
$\displaystyle \int\frac{f^{\prime}(x)}{f(x)}\,dx=\ln|f(x)|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The accumulation function’s derivative
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$