Evaluating Logarithmic Functions
Analyze the equation of the function.
The base of the logarithm is 5. So, the values of $x$ represent powers of 5. The values of $f(x)$ represent the exponents of those powers.
Choose several values of $x$ that are wholenumber powers of 5.
Since $5^0=1$, start with 1. Then keep multiplying by 5 to generate other powers of 5.
$x$ 
Explanation 

$1$ 
$5^{0}=1$ 
$5$ 
$5^{1}=5$ 
$25$ 
$5^{2}=25$ 
$125$ 
$5^{3}=125$ 
$625$ 
$5^{4}=625$ 
Use the relationship between logarithms and exponents to identify the corresponding values of $f(x)$.
$x$ 
$f(x)=\log_5{x}$ 
Explanation 

$1$ 
$0$ 
The logarithmic form of $5^{0}=1$ is $\log_{5}1=0$. 
$5$ 
$1$ 
The logarithmic form of $5^{1}=5$ is $\log_{5}5=1$. 
$25$ 
$2$ 
The logarithmic form of $5^{2}=25$ is $\log_{5}25=2$. 
$125$ 
$3$ 
The logarithmic form of $5^{3}=125$ is $\log_{5}125=3$. 
$625$ 
$4$ 
The logarithmic form of $5^{4}=625$ is $\log_{5}625=4$. 
Analyzing the Base of a Logarithmic Function
Evaluate the first function to identify several ordered pairs on the graph.
$x$ 
$f(x)=\log_{2}x$ 
Explanation 

$0.25$ 
$2$ 
The logarithmic form of $2^{2}=0.25$ is $\log_{2}0.25=2$. 
$0.5$ 
$1$ 
The logarithmic form of $2^{1}=0.5$ is $\log_{2}0.5=1$. 
$2$ 
$1$ 
The logarithmic form of $2^{1}=2$ is $\log_{2}2=1$. 
$4$ 
$2$ 
The logarithmic form of $2^{2}=4$ is $\log_2{4}=2$. 
$x$ 
$f(x)=\log_{\footnotesize{\frac{1}{2}}}{x}$ 

$0.25$ 
$2$ 
$0.5$ 
$1$ 
$2$ 
$1$ 
$4$ 
$2$ 
Determine the $x$intercept and the asymptote.
A logarithmic function in the form $f(x)=log_{b}x$ has an $x$intercept of $(1, 0)$ and the $y$axis as the vertical asymptote. Therefore, the logarithmic function has an $x$intercept at $(1, 0)$ with the $y$axis as the vertical asymptote.
Plot the $x$intercept and the ordered pairs. Connect them with continuous curves. The first function, $f(x)=\log_{2}x$ is increasing.The second function, $f(x)=\log_{\footnotesize{\frac{1}{2}}}x$ is decreasing.
Transformations of Logarithmic Functions
Translations of Logarithmic Functions
Vertical Translations  Horizontal Translations 

For the graph of $f(x)=\log_b{x}$ and $k>0$:

For the graph of $f(x)=\log_b{x}$ and $h>0$:

Stretches, Compressions, and Reflections of Logarithmic Functions
Stretches and Compressions  Reflections 

For the graph of $f(x)=\log_b{x}$ and $a>0$:

For the graph of $f(x)=\log_b{x}$:

Transformations of $y=\log_2{x}$
$f(x)=\log_2{(x+1)}$  $f(x)=\log_2{x}$  $f(x)=\log_2{x}3$ 

The graph shifts left 1 unit.  The graph reflects across the $x$axis.  The graph shifts down 3 units. 
Identify the parent function.
The base in the logarithm in the given function is 2. So, the parent function is the simplest logarithmic function with base 2.
Determine whether there are any translations.
 Since $h=1$, there is a translation 1 unit to the left.
 Since $k=3$, there is a translation 3 units down.