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

How do you solve a logarithmic equation with logs on both sides?

​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 1: Simplify, then divide both sides by 2 to get x 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.

When the same operation is on both sides of the equation, they cancel out.

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

How to solve an equation with logarithms.

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.

How to solve a logarithmic equation with logs on both sides.

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!

When you have the same operation (the same type of logarithm), they cancel 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.

log base 9 of 3x is equal to log base 9 of 15
When both sides have the same operation, they cancel out!

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