This is the last section that falls under logarithms and can be either easy or hard depending on your grasp of logarithms. This is because to solve these problems, you must use everything you have learnt about logarithms from something simple to something more complex.

To explain our objectives more clearly, we must turn our logarithms into their base form to then solve for X.

This is an easy example: Log3(x)=4

we will start by using the components to turn it back into the exponential form of 3^4=x.

From there, it is a matter of plugging the numbers and then get the value for X which is 81=x

For a medium level example, we can use Log2(x+3)=5.

We will once again turn it into exponential form which is 2^5=x+3

We solve for the 2^5 to turn this into 32=x+3 which we will then subtract 3 to isolate the x.

32-3=x will then give us our final answer which is 29=x

For the more challenging level, we will use Log3(x)+Log3(x-8)=4

This shows us that we have to use the product rule to condense the logarithm into Log3(x(x-8))=4

Now yes we can turn it into exponential form which is x(x-8)=3^4

simplifying the equation we will get x(x-8)=81

However, we are not finished since we have the distributive law to use to simplify the equation into x^2-8x=81

If you notice, we have the conditions to turn this into a quadratic which we can solve by quadratic formula or factorization methods. We take x^2-8x-81 and we must choose what method to use. The only option however would be to use the quadratic formula which is: -b(+or-)the square root ofb^2-4(ac) all over 2a. plotting our equation into the formula gives us 

8(+or-) square root of 64-4(1*-81) all over 2.

We plug that into our calculators to recieve answers 27.7/2 or -11.7/2. We of course must always choose the positive answer which means that X is approximately 13.85(2d.p.)

Logarithmic equations can be one of the subtopics under logarithms that can cause the most confusion without a doubt. For this, you need to solve the problems by remembering not only logarithm rules but many more concepts like distributive laws or how to solve for quadriatics.