The logarithmic function

Exponential and logarithmic functions, grade 12 (5 units) Lesson 9 of 23Learning

The logarithmic function

The inverse function of

\displaystyle a^x

The function $\displaystyle {y}={\log_a x}$ is the inverse function of $\displaystyle {y}={a^x}$ : it takes the result back to the exponent.

$\displaystyle {2^3}={8}$ , so $\displaystyle {\log_2 8}={3}$ : the point $\displaystyle (3,8)$ is on the graph of $\displaystyle 2^x$ , and the point $\displaystyle (8,3)$ on the graph of $\displaystyle \log_2 x$ .
So at every point: $\displaystyle (x,y)$ is on the graph of $\displaystyle a^x$ exactly when $\displaystyle (y,x)$ is on the graph of $\displaystyle \log_a x$ .

\displaystyle f(x)=2^{x}

\displaystyle g(x)=\log_{2}x

\displaystyle y=x

\displaystyle A=(1,2)

\displaystyle B=(2,1)

Swapping $\displaystyle x$ and $\displaystyle y$ is a reflection in the line $\displaystyle {y}={x}$ : the point $\displaystyle A(1,2)$ goes to $\displaystyle B(2,1)$ .
So the graph of $\displaystyle \log_2 x$ is the reflection of the graph of $\displaystyle 2^x$ in the line $\displaystyle {y}={x}$ .

The properties of

\displaystyle \log_a x

Every property of $\displaystyle a^x$ passes, reflected, to a property of $\displaystyle \log_a x$ ( $\displaystyle a>0$ , $\displaystyle a\neq1$ ):

The domain: $\displaystyle x>0$ – because $\displaystyle a^x$ takes only positive values.
The function takes every value $\displaystyle y$ .
The graph passes through $\displaystyle (1,0)$ , because $\displaystyle {\log_a1}={0}$ , and through $\displaystyle (a,1)$ , because $\displaystyle {\log_a a}={1}$ .
The vertical asymptote: $\displaystyle {x}={0}$ – the reflection of the horizontal asymptote $\displaystyle {y}={0}$ of $\displaystyle a^x$ .
When $\displaystyle a>1$ the function increases; when $\displaystyle 0<a<1$ it decreases.

A reciprocal base:

\displaystyle \log_{\frac{1}{a}}x

By the change of base, $\displaystyle {\log_{\frac{1}{a}}x}={\frac{\log_a x}{\log_a\frac{1}{a}}}={\frac{\log_a x}{-1}}$ :

\displaystyle \log_{\frac{1}{a}}x

\displaystyle =

\displaystyle -\log_{a}x

\displaystyle f(x)=\log_{2}x

\displaystyle g(x)=\log_{\frac{1}{2}}x

\displaystyle h(x)=\log_{3}x

\displaystyle k(x)=\log_{\frac{1}{3}}x

The graph of $\displaystyle \log_{\frac{1}{2}}x$ is the reflection of the graph of $\displaystyle \log_2 x$ in the x-axis; so are $\displaystyle \log_{\frac{1}{3}}x$ and $\displaystyle \log_3 x$ (dashed).
All four graphs pass through $\displaystyle (1,0)$ .
$\displaystyle \log_3 x$ rises more slowly than $\displaystyle \log_2 x$ when $\displaystyle x>1$ : $\displaystyle {\log_2 8}={3}$ , but $\displaystyle \log_3 8<2$ .

The domain of

\displaystyle \log_a f(x)

$\displaystyle \log_a f(x)$ is defined only where $\displaystyle f(x)>0$ . To find the domain, solve the inequality $\displaystyle f(x)>0$ .

$\displaystyle \log_3(5-x)$ : $\displaystyle 5-x>0$ , that is $\displaystyle x<5$ .
$\displaystyle \log_2(x^2-4x)$ : $\displaystyle x^2-4x>0$ , that is $\displaystyle x(x-4)>0$ – $\displaystyle x<0$ or $\displaystyle x>4$ .
$\displaystyle \log_2(x^2+1)$ : $\displaystyle x^2+1>0$ for every $\displaystyle x$ , so the function is defined for every $\displaystyle x$ .

Transformations of

\displaystyle \log_a x

The example below investigates two transformations of $\displaystyle \log_2 x$ :

$\displaystyle {f(x)}={\log_2(x-3)+1}$ : a shift 3 to the right and 1 up. The domain is $\displaystyle x>3$ , and the vertical asymptote $\displaystyle {x}={3}$ .
The point on the x-axis: $\displaystyle {\log_2(x-3)}={-1}$ , that is $\displaystyle {x-3}={\frac{1}{2}}$ and $\displaystyle {x}={3.5}$ . The graph does not meet the y-axis, because $\displaystyle {x}={0}$ is not in the domain.
$\displaystyle {g(x)}={\log_2(-x)}$ : a reflection of $\displaystyle \log_2 x$ in the y-axis. The domain is $\displaystyle x<0$ , and the graph passes through $\displaystyle (-1,0)$ .

\displaystyle f(x)=\log_{2}(x-3)+1

\displaystyle x=3

\displaystyle g(x)=\log_{2}(-x)

\displaystyle x=0

Exercise 1

The function $\displaystyle {f(x)}={\log_3(x+2)-1}$ is given.

Find the domain of the function: write an inequality and solve it.

Find the point where the graph meets the x-axis: solve the equation.

Write the equation of the vertical asymptote, and mark on the graph the points where it meets the axes.

\displaystyle f(x)=\log_{3}(x+2)-1

Exercise 2

Find the domain: write an inequality and solve it.

$\displaystyle {f(x)}={\log_2(x^2-9)}$

$\displaystyle {g(x)}={\log(4x-x^2)}$

Exercise 3

The graph in the drawing is the graph of a function of the form $\displaystyle {y}={\log_a x}$ .
The graph passes through the point $\displaystyle (9,2)$ .
Find $\displaystyle a$ , and write the function.

\displaystyle y=\,?

\displaystyle (9,2)

Exercise 4

The function $\displaystyle {f(x)}={\log_{\frac{1}{2}}x}$ is given.

Does the function increase or decrease? Explain with $\displaystyle {\log_{\frac{1}{2}}x}={-\log_2 x}$ .

Solve the inequality $\displaystyle f(x)>-3$ .

Exercise 5

The functions $\displaystyle {f(x)}={\log_2 x}$ and $\displaystyle {g(x)}={\log_2(8-x)}$ are given.

Find the x-coordinate of the point where the graphs meet: solve the equation.

Mark the meeting point on the graph.

\displaystyle f(x)=\log_{2}x

\displaystyle g(x)=\log_{2}(8-x)

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
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$