A Nice Algebra Problem | Find a | Olympiad Mathematics

[00:00:00]Hello everyone, you are welcome. Today we have a very interesting math question.

[00:00:06]Here is 20 raised to power A plus 20 raised to power A plus 20 raised to power A divided by 10 raised to power A plus 10 raised to power A plus 10 raised to power A is equal to 100.

[00:00:18]So we will try to find out the value of A.

[00:00:21]So let's start our solution. First of all, we can here look into the left-hand side. Here in the numerator, we can take out this 20 raised to power A common.

[00:00:31]But here this same number, same expression is added three times.

[00:00:34]Therefore, here we can write this expression in the numerator as this is just three times 20 raised to power A.

[00:00:46]Similarly, in the denominator here this 10 raised to power A is added three times. So therefore, again here we can write this as three times 10 raised to power A.

[00:00:58]is equal to 100.

[00:01:02]Here in the left-hand side, we can cancel three with three.

[00:01:06]So this will become only this one expression, 20 raised to power A divided by 10 raised to power A.

[00:01:12]We look at the both the expression, both expression has the same power.

[00:01:16]So here in this step, we will use the same power exponential math property.

[00:01:20]So here we will use this one exponential math property. We can write X raised to power A divided by Y raised to power A as X by Y whole raised to power A.

[00:01:37]So using this exponential property, here this left-hand side will become this will become 20 by 10 raised to the power A is equal to 100.

[00:01:56]Here we will cancel zero with zero. So, this will become two by one. And two by one it is simply two. So, this will become two raised to power A is equal to 100.

[00:02:09]How can we solve this equation further to find out the value of A?

[00:02:14]Here say we can take common log on both sides.

[00:02:17]So, here we'll take common log on both sides and we'll find out the value of A.

[00:02:23]This will become log of two raised to power A is equal to log of This is 100. So, we can write this 100 as 10 squared.

[00:02:37]And in both sides of this equation we will use a logarithm property.

[00:02:42]So, in both sides we will use this one log property. We can write log of A raised to power X as X times log of A.

[00:02:57]So, using this log property here in both sides we will move the powers to the front of the log numbers.

[00:03:05]So, this will become This will become A times log of two is equal to and this will become two times log of 10.

[00:03:19]We will divide both sides by log of two to find out the value of A.

[00:03:24]So, this will become log of two.

[00:03:27]And here we'll also divide this expression by log of two.

[00:03:32]And in left hand side this log of two and this log of two will be cancelled.

[00:03:36]So, this will become only A is equal to and this will become two times log of 10 divided by log of two.

[00:03:51]Here we can replace this log 10 having base 10 with 1 because this is a common log. And we also know that log of any number with the same base is always 1.

[00:04:02]But here we will leave this number as the same to further simplify this log number in the denominator.

[00:04:07]So here in this expression we will use change of base logarithm math property.

[00:04:12]So here in this expression we will use this one log property.

[00:04:15]We can write log of x divided by log of y this is log of x with base y.

[00:04:28]So using this change of base logarithm math property here this will become a is equal to this will become two times log of 10 with base two.

[00:04:44]And this is approximately equal to 6.64.

[00:04:51]So this is the final answer and final value of a.

[00:04:55]And we'll try to verify and check that is a is equal to this value or this value verify the above equation or not.

[00:05:05]So let's try to verify this value.

[00:05:08]So to verify the value here we will rewrite our question again.

[00:05:13]But if we substitute the value of a here in this equation so it will be a little bit different and a little bit longer method.

[00:05:20]Here we will write the simplified equation of this one question that is 2 raised to power a is equal to 100.

[00:05:29]So this will become simplified equation is 2 raised to power a is equal to 100.

[00:05:39]Here we substitute the value of a so this become 2 raised to power the value of a is two times log of 10 with base two is equal to 100.

[00:05:56]Now here in the left-hand side we will use an explanation math property.

[00:06:00]So in the left-hand side we will use this one explanation math property. We know that we can write A raised to power M times N as A raised to power N whole raised to power M.

[00:06:13]So using this explanation math property here this expression in the left-hand side will become this will become two raised to power log of 10 with base two whole square is equal to 100.

[00:06:32]Here inside the parentheses we will use another log property.

[00:06:37]We know that we can write A raised to power log of X with base A as X.

[00:06:46]So using this log property here we will replace this left-hand side with only this one number that is 10.

[00:06:54]So the left-hand side will become this will become 10 square is equal to 100.

[00:07:02]And 10 square is 100.

[00:07:06]Is equal to 100.

[00:07:09]Since both sides are equal so it's mean that this value of A can verify the above interesting explanation math question.