Solving Equations with Logarithms on Both Sides
Let's start with an equation without any logarithms: 2(x) = 2(6)
Do you know the answer? You could simplify both sides and get 2x = 12. Then divide both sides by 2 and you see that x = 6.
Can you look back now and figure out the answer without doing much work? 2 times some mystery number is the same as 2 times 6. What must the mystery number be? 6. When you have the same operation on each side, they essentially cancel each other out.
Now let's try an equation with logarithms: log x = log 100
Don't let this scare you off just because you see logarithms. You could simplify log 100 and rewrite it as a 2 (It's a common log with base 10 and 10 to the second power is 100). So that would give you log x = 2. Then you could rewrite that as an exponential equation 10 squared equals x, which would give you x = 100. But that's a whole lot of unnecessary work when instead, you can just look at the original equation and see that the logs just cancel each other out. (If all of this logarithm stuff sounds completely foreign to you, you might want to start at the intro to logarithms lesson before moving on.)
Do you know the answer? You could simplify both sides and get 2x = 12. Then divide both sides by 2 and you see that x = 6.
Can you look back now and figure out the answer without doing much work? 2 times some mystery number is the same as 2 times 6. What must the mystery number be? 6. When you have the same operation on each side, they essentially cancel each other out.
Now let's try an equation with logarithms: log x = log 100
Don't let this scare you off just because you see logarithms. You could simplify log 100 and rewrite it as a 2 (It's a common log with base 10 and 10 to the second power is 100). So that would give you log x = 2. Then you could rewrite that as an exponential equation 10 squared equals x, which would give you x = 100. But that's a whole lot of unnecessary work when instead, you can just look at the original equation and see that the logs just cancel each other out. (If all of this logarithm stuff sounds completely foreign to you, you might want to start at the intro to logarithms lesson before moving on.)
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 out.
Example 1:
First, make sure the logs on both sides have the same base. These both have base 7, so they cancel each other out.
Example 2:
Once again, start by checking to make sure the logarithms on both sides have the same base. Both logs have base 9, which means they cancel out.
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Want to learn how to solve a different type of logarithmic equation?
Check out the lesson on solving logarithmic equations with logarithms on one side.
