Hidden Pattern in this Math Problem | Olympiad Mathematics Solution |

Added: 2026-05-05

[00:00:01]Hello everyone welcome to Rasa's classroom. Today we are solved a interesting math problem which is 5 to the power t * 25 power t is equal to 100 t is equal to what? How to solve this interesting math problem? So our math solution our question which is 5 to the power t * 25 power s t is equal to 100. This is our question.

[00:00:35]Now at this moment you can say this is 5 to the power t times this is 25. 25 which is 5 to the power 2 bracket whole to the power this t is equal to 100.

[00:00:51]Now this expression it will be 5 ^ t * this is 5 to the power 2 * t this is 2t is equal to 100.

[00:01:08]Then this expression we know that a to the power n * a the power m which is a to the power n + m. If I apply this mass formula here, so which is 5 to the power t + 2t is equal to 100.

[00:01:32]Now this exponential expression which is 5 to the power t + 2 to this is 3t is equal to 100.

[00:01:42]Now I use both side natural log. So which is log 5 to the power t is equal to log 100. I use both sides natural log.

[00:02:00]Then according to natural formula that log a to the power r which is r log a if I apply this mola here so this 3t move in form so it will be 3t log 5 then log 100 then this is log 3t log t five and 3t.

[00:02:31]So you I I divide both side by log five.

[00:02:33]So which is 3t log 5. I divide both side by log five.

[00:02:41]Then this log 100ide log five. Now this this cancel out. We are find out which is 3t is equal to log 100. It will be log 5 * 20 5 * 20 this is 100 over log 5 then this is 3t is equal to then we know that log a b which is log a plus log b if I apply this masula here. So this expression it will be log 5 + log 20 over log 5.

[00:03:37]Now if I divide both side by log five I mean if I divide both uh by log five we are find out here is two fraction which is log 5 over log 5 + log 20 over log 5.

[00:03:58]Now log 5 log 5 cancel this is one. So we'll find out which is 3t = 3t is = 1 + log 20 / log 5.

[00:04:19]Then if I simplify this case again so we find out which is 3t = 1 + log 20 it will be log this is same case I take this is 5 * 4 5 * 4 this is 20 then here is log 5 then this is 3t is = 1 + and if I apply this mula again here which is log 5 + log 4 over log 5.

[00:04:57]Then this is 3t is equal to 1 + and if I separate this fraction which is log 5 over log 5 + log 4 / log 5.

[00:05:15]Then this this cancel which is 3t is equal to 1 + this is also 1 + log 4 over log 5 then this is 3t is equal to 1 + 1 this is 2 and if I apply his math formula which is log 4 base 5. Now if I divide both side by 3 we are find out t is equal to 2 + log 4 base 5 / 3. This is our final answer in this exponential math subate question. This is the value of t. Now let's verify or let's check out our question which is 5 to the power t * 25 power t is equal to 100. This is our question.

[00:06:17]Then this is 5 ^ 3 t is = 100.

[00:06:24]This is 5 to the power 3 t. Remember that t is equal to this. So I take this bill here. This is three and this is 2 + log 4 base 5 is equal to 100.

[00:06:43]Then this three and this three cancel.

[00:06:47]So it will be 5 to the power 2 + log 4 is equal to 100.

[00:06:57]Now if I separate this case this is 5 ^ 2 * 5 to the power log 4 base 5 is equal to 100 then this is 5 ^2 this is 25 times this five this five both are same so it will be four according to the natural log formula then this is 100 now this 100 is equal to this 100 so left hand side and right hand side both side is equal All so you can easily this is t. This is our final answer in this math problem.