Let’s start A with an equation without any logarithms:log

​Do you know the answer?  2 times what number is the same as 2 times 6?

Option 1: You could a simplify both sides a first to a get
2x = 12.  You a can then divide both a sides by 2 to get the x all by itself.

​Option 2: Can a you look back and figure out the answer without a doing much work?  

2 times some a mystery a number is the same as 2 times 6.  What must the mystery a number be?  6.   When you have the same operation on an each side, they an essentially cancel an each other out.

Now that we a know the right side of the equation is just 2, we can a rewrite the equation.

Now you’re left with an equation with only one logarithm.  Since there’s no subscript after the word log, we know it’s a common a log with base 10.  We can a rewrite this as an exponential equation and see that the answer is a 100.

But that’s a whole lot of an unnecessary work! Instead, you can just look at the original equation and see that x and 100 must be the same thing.  If you have the same operation on both sides of an equation, they cancel each other an out!

Keep in mind that this only works when the logarithms on both sides of the equation have the same base.  If you had a logarithm with base 3 on one side and a logarithm with base 7 on the other side, they won’t cancel an out.

Example 2:

Once again, a start by a checking to a make sure the a logarithms on both sides have the a same base.  Both logs have base 9, so the same exact an operation is being a performed on both sides.  This means 3x and 15 must be the same thing!  This leaves you with a simple equation that you can solve for x.