Logarithm Functions

Logarithm Functions

Unit 1 Methods

Warm-ups

What do we know?

Multiple Choice

Simplify: $\frac{a^2b}{\left(2ab^2\right)^3}\div\frac{ab}{16a^0}$  

•  $\frac{2}{a^2b^6}$  

•  $\frac{2a^2}{b^6}$  

•  $2a^2b^6$  

•  $\frac{1}{128ab^5}$  

Multiple Choice

The function $f:\ R\rightarrow R,\ f\left(x\right)=3\times2^x-1$  has the range:

•  $R$  

•  $R$  \ $\left\{1\right\}$  

•  $\left(-1,\ \infty\right)$  

•  $\left(1,\ \infty\right)$  

• [ $1,\infty$  )

Multiple Choice

Which of the following graphs could be the graph of the function $f\left(x\right)=2^{ax}+b$  

Multiple Choice

Express  $\log_{10}4+2\log_{10}3-\log_{10}6$  as a single logarithm

•  $\log_{10}4$  

•  $\log_{10}5$  

•  $\log_{10}-6$  

•  $\log_{10}6$  

Multiple Choice

 $a^x=b$  and  $\log_ab=x$  are equivalent

• True

• False

13G Graphs of Logarithm Functions

Unit 1 Methods

Graph of $y=\log_ax$  when a>1

What do you notice about the graph of  $y=2x$  and  $y=\log_2x$  ?

The graph of $y=\log_2x$  is the reflection of the graph  $y=2^x$   in the line  $y=x$  

Multiple Choice

What is the domain and range of the following:

• R, R

• R+, R

• R-, R

Multiple Choice

What is the inverse of the following  $f\left(x\right)=2^x$ ?

•  $g\left(x\right)=\log_2x$  

•  $g\left(x\right)=\log_2y$  

We know that a graph with a negative exponent (or a base 0<a<1) is a decay.

Logarithms for these functions are also a decay reflected over the y=x.

13F Using logarithms to solve equations

Consider the equation  $2^x=11$  

We could rewrite this as:
 $\log_{10}2^x=\log_{10}11$  
so:
 $x=\frac{\log_{10}11}{\log_{10}2}$  
but from the original equation  $2^x=11$  , so
 $\log_211=\frac{\log_{10}11}{\log_{10}2}$  

 $\log_bc=\frac{\log_ac}{\log_ab}$