Logarithms and logarithmic functions have been thoroughly covered in:
This section provides an inanutshell, ataglance summary of key results.What is $\,\log_b x\,$? 
Logarithms are exponents! The number ‘$\,\log_b x\,$’ is the power (the exponent) that $\,b\,$ must be raised to, in order to get $\,x\,$. $$\log_b x = y\ \ \iff\ \ b^y = x$$ Read ‘$\,\log_b x\,$’ as ‘log base $\,b\,$ of $\,x\,$’ . The equation ‘$\,\log_b x = y\,$’ is called the logarithmic form of the equation. The equation ‘$\,b^y = x\,$’ is called the exponential form of the equation. 
$\log_2 8 = 3\,$ since $\,2^3 = 8\,$ $\log_7 1 = 0\,$ since $\,7^0 = 1\,$ $\log_3 \frac 13 = 1\,$ since $\,3^{1} = \frac 13\,$ 
Allowable Bases for Logs 
In the expression ‘$\,\log_b x\,$’ , the number $\,b\,$ is called the base of the logarithm. The number $\,b\,$ must be positive and not equal to $\,1\,$: $\,b > 0\,$ and $\,b\ne 1\,$ 
allowable bases for logarithms: $b > 0\,$, $\,b\ne 1$ 
Two Special Logarithms 
The function ‘$\,\log_{10}\,$’ (log base $\,10\,$) is called the common logarithm. It is often abbreviated as just ‘$\,\log\,$’ (with no indicated base). The function ‘$\,\log_{\text{e}}\,$’ (log base $\,\text{e}\,$) is called the natural logarithm. It is often abbreviated as ‘$\,\ln$’ (with no indicated base). Caution: Some disciplines use ‘$\,\log\,$’ to mean the natural logarithm. Always check notation. 
$\log x$ the common log of $\,x$ $\ln x$ the natural log of $\,x$ 
Allowable Inputs for Logs 
In the expression ‘$\,\log_b x\,$’ , the number $\,x\,$ (the input) must be positive: $\,x > 0\,$ 
allowable inputs for logarithms: $x > 0$ 
Function View of Logs 
The number ‘$\log_b x\,$’ is the output from the function ‘$\,\log_b\,$’ when the input is ‘$\,x\,$’. The domain of the function $\,\log_b\,$ is the set of all positive real numbers: $\,\text{dom}(\log_b) = (0,\infty)$ The range of the function $\,\log_b\,$ is the set of all real numbers: $\,\text{ran}(\log_b) = \Bbb R$ 
function view of logarithms 
Laws of Logarithms 
Let
$\,b\gt 0\,$,
$\,b\ne 1\,$,
$\,x\gt 0\,$, and
$\,y\gt 0\,$.
$\log_b\,xy = \log_b x + \log_b y$
$\displaystyle \log_b\frac{x}{y} = \log_b x  \log_b y$
For this final property, $\,y\,$ can be any real number: 
See Properties of Logarithms for a typical proof of these laws. 
Change of Base Formula for Logarithms 
Let $\,a\,$ and $\,b\,$
be positive numbers that are not equal to $\,1\,$, and let $\,x\gt 0\,$. Then, $$ \log_b\,x =\frac{\log_a\,x}{\log_a\,b} $$ In words: You can change from any base $\,b\,$ to any base $\,a\,$; the ‘adjustment’ is that you must divide by the log to the new base ($\,a\,$) of the old base ($\,b\,$). 
See Change of Base Formula for Logarithms for a derivation of this formula. The equation $$\,\log_b x = \left(\frac 1{\log_a b}\right)(\log_a x)\,$$ shows that any log curve is just a vertical scaling of any other log curve! 
Logarithm functions are onetoone 
Since logarithms are functions: $x = y \ \ \Rightarrow\ \ \log_b x = \log_b y$ When inputs are the same, outputs are the same. Since logarithms are onetoone: $\log_b x = \log_b y\ \ \Rightarrow\ \ x = y$ When outputs are the same, inputs are the same. Thus, for all $\,b > 0\,$, $\,b\ne 1\,$, $\,x > 0\,$ and $\,y > 0\,$: $$x = y\ \ \iff\ \ \log_b x = \log_b y$$ 

Inverse Properties 
Logarithmic functions are
onetoone,
hence have
inverses. The inverse of the logarithmic function with base $\,b\,$ is the exponential function with base $\,b\,$. For $\,b > 0\,$, $\,b\ne 1\,$, and all real numbers $\,x\,$: $$\log_b b^x = x$$ For $\,b > 0\,$, $\,b \ne 1\,$, and $\,x > 0\,$: $$b^{\log_b x} = x$$ 

Special Points 
For $\,b > 0\,$ and $\,b\ne 1\,$:

Graphs of Logarithmic Functions
Properties shared by all logarithmic graphs, $\,y = \log_b x\,$:

For $\,b > 1\,$:

$\,b > 1\,$: blue curve: $\,y = \log_b x\,$ red curve: inverse $\,y = b^x\,$ dashed line: $\,y = x\,$ 
For $\,0 < b < 1\,$:

$\,0 < b < 1\,$: blue curve: $\,y = \log_b x\,$ red curve: inverse $\,y = b^x\,$ dashed line: $\,y = x\,$ 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
