Following all the rules listed in our introduction,  we can either expand or condense our logarithms. For this section, we will only look at how to condense our logarithms.

For starters, lets take the equation log(5)+log(20), we will use the product rule to condense it and turn it into Log(5*20) or Log(100).

the same applies when we have equations where we subtract logarithms.

However, logarithmic equations can get much longer and involve many more steps like this one: 

2Log(4)+5log(8)-Log(3).

Following our order of operations, we will be using the power rule first which shortens the equation just a bit into:

Log(4^2)+Log(8^5)-Log(3).

next in our order of operations, it is addition and subtraction as they appear from left to right. Following this rule means we use the product rule next.  Log(4^2 * 8^5)-Log(3).

last but not least, we use the quotient rule to turn this into a single logarithm which is: Log(4^2 *8^5 all over 3) or Log(174762.67(2d.p.))

note that we can use all the rules we have at our disposal depending on the problem we are facing.

To begin with, condensing logarithms isn't too difficult to understand once you have learnt to grasp the different rules you need to follow to condense these equations. Both Exponential and logarithmic equations can easily pose more trouble given the fact that you need to keep track of many more things and solve for a variable in particular. These however don't require as many examples to understand the methods used to solve them.