Graphs of Logarithmic Functions (Ex 13G)

Year 11 Mathematical Methods Unit 1 · Cambridge §13G · Mr Wong

Name

Class

Date

Learning intentions.

Sketch $y=\log_a(x)$ as the reflection of $y=a^x$ in the line $y=x$
State domain, range, asymptote and $x$-intercept of transformed log graphs
Find the inverse of an exponential or a log function and state its domain and range

Key results

$y=\log_a(x)$ ($a>1$): dom $x>0$, range $\mathbb{R}$, asymp $x=0$, $x$-int $(1,0)$

$y=\log_a(x-h)+k$: dom $x>h$, asymp $x=h$, $x$-int where $x-h=a^{-k}$

$\log_{1/a}(x)=-\log_a(x)$ (reflection in $x$-axis)

Part A — Translations of $y=\log_2(x)$

1. For each graph, state the domain, asymptote and $x$-intercept.

a $y=\log_2(x)$ dom: asy: $x$-int:

b $y=\log_2(x-2)$ dom: asy: $x$-int:

c $y=\log_2(x+5)$ dom: asy: $x$-int:

d $y=\log_2(x)+1$ dom: asy: $x$-int:

Part B — Reflections

2. For each, state the domain, asymptote and $x$-intercept, and say whether the curve is increasing or decreasing.

a $y=-\log_2(x)$

b $y=\log_2(-x)$

c $y=\log_{1/2}(x)$

3. Show, using change of base, that $\log_{1/2}(x)=-\log_2(x)$.

Part C — Combined transformations

4. Sketch $y=\log_2(3x-2)$. Mark the asymptote, $x$-intercept and one other point. State the domain and range.

y x

5. Sketch $y=\log_{1/2}(x+1)-2$. Mark the asymptote, both axis intercepts. State the domain and range.

y x

Log Graphs 13G — continued

Part D (find the inverse) · Challenge

Name

Part D — Inverse functions

Swap $x$ and $y$, solve for $y$, then write the domain and range of $f^{-1}$.

6. $f:\mathbb{R}\to\mathbb{R}$, $f(x)=2^x-5$. $f^{-1}(x)=$ dom: range:

7. $f:\mathbb{R}\to\mathbb{R}$, $f(x)=2\cdot 5^x+2$. $f^{-1}(x)=$ dom: range:

8. $f:(-3,\infty)\to\mathbb{R}$, $f(x)=\log_2(x+3)-2$. $f^{-1}(x)=$ dom: range:

9. $f:\mathbb{R}^+\to\mathbb{R}$, $f(x)=\log_3(x)$. $f^{-1}(x)=$ dom: range:

Challenge

10. On the same axes, sketch $y=2^x$, $y=\log_2(x)$ and $y=x$. Show that $y=2^x$ lies entirely above the line $y=x$, and (by reflection in $y=x$) $y=\log_2(x)$ lies entirely below $y=x$. Hence neither curve crosses $y=x$ — so they don't intersect each other either.