Logarithmic functions

Logarithmic functions > Logarithmic functions

Overview

12345Logarithmic functions

Exercises

Exercise 1

Construct the graph of the function $f (x) = 1 - 3 \cdot log (x + 4)$ .

Write down the domain and range of $f$ .

Write down the equation of the vertical asymptote.

What transformations of $y = log (x)$ result in the graph of $f$ ?

Algebraically determine the `x` -intersect of the graph of $f$ .

Exercise 2

The graphs of the functions $f (x) = {(\frac{1}{2})}^{x}$ and $g (x) = 2^{x}$ are mirror images of each other with respect to the $y$ -axis.
The graphs of the functions $h (x) = {}^{\frac{1}{2}} log (x)$ and $k (x) = {}^{2} log (x)$ therefore must be mirror images of each other with respect to the $x$ -axis. That means that $h (x) = - k (x)$ .

Which value of $x$ results in $h (x) = 3$ ?

Which value of $x$ results in $k (x) = -3$ ?

The point $(\frac{1}{8}, 3)$ on the graph of $h$ has a mirror image on the graph of $k$ . What are the coordinates of this mirror image?

Choose coordinates of another point on the graph of $h$ and find the coordinates of the corresponding mirror image on the graph of $k$ .

Plot the graphs of $h$ en $k$ in the same coordinate system and solve: $h (x) = k (x)$ .

Now show that $h (x) = - k (x)$ for any $x > 0$ . To do so, write down both function rules in a format that you can enter in your graphing calculator.

Exercise 3

Light sensitivity of photographic material is expressed as a sensitivity index. The most common system for this is the ASA-systeem (American Standards Association). On rolls of film you often also find a second sensitivity indicator, the DIN-value. The relationship between ASA and DIN is given by the formula

$y = 1 + a \cdot log x$

In the formula $x$ is light sensitivity in ASA and $y$ is light sensitivity in DIN. A film of $100$ ASA has a DIN value of 21.

Determine $a$ .

Draw the graph. Films usually have ASA values between $50$ and 1000.

What is the ASA index for a film with a light sensitivity of $31$ DIN?

Exercise 4

Given the functions $f (x) = {}^{2} log (x)$ and $g (x) = {}^{2} log (2 - x)$ .

Determine the domain, the range and the asymptote of both $f$ and $g$ .

The graph of function $g$ is derived by transforming the graph of $f$ . Describe the necessary transformations in the correct order.

Draw the graphs of the functions $f$ and $g$ and solve: $f (x) = g (x)$ .

Across which line are the graphs of $f$ and $g$ mirror images of each other?

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