Solve for x in this nice Algebra Problem | x=? | Olympiad Mathematics

Added: 2026-06-03

[00:00:00]Hello everyone, you are welcome. Today we have a very beautiful exponential math problem.

[00:00:07]What is the value of x in this beautiful math problem which is 5 raised to power x by 25 is equal to 50?

[00:00:16]So let's start our solution. First of all here we can write both sides of this equation as this is 5 raised to power x divided by 25. 25 can be written as this is 5 squared is equal to and here this 50 this is simply 25 * 2. So here we can write this 25 as 5 squared * 2.

[00:00:44]Then we will take this 5 squared to the left hand side.

[00:00:47]So this will be divided here.

[00:00:50]So this will become 5 raised to power x divided by this will become 5 squared * 5 squared is equal to 2.

[00:01:03]Here in the denominator this base 5 is same so we'll add the powers.

[00:01:09]So this will become 5 raised to power x divided by 5 raised to power 2 + 2 this is 4.

[00:01:19]is equal to 2.

[00:01:22]Here in the left hand side look at to the numerator and denominator both have same base and different powers.

[00:01:29]So here we will use this one exponential math property.

[00:01:32]We can write a raised to power m divided by a raised to power n as a raised to power m - n.

[00:01:43]So using this exponential property here this equation will become it will become 5 raised to power x - 4 is equal to 2.

[00:01:56]Now this is an exponential equation. So, here we will take a log on both sides.

[00:02:02]So, this will become log of 5 raised to power x minus 4 is equal to log of two.

[00:02:13]And in the left hand side, we will use an exponential log property and we will move the power to the front of log.

[00:02:20]So, this will become this is x minus 4 times log of 5 is equal to log of two.

[00:02:33]We will divide both sides by this number, log of 5.

[00:02:42]So, here in the left hand side, this log of 5 and this log of 5 will be canceled.

[00:02:47]So, this will become only x minus 4.

[00:02:49]And here in the right hand side, we will use change of base logarithm property.

[00:02:54]So, here in the right hand side, we will use this one log property. We can write log of x divided by log of a as log of x to base a.

[00:03:11]So, using this change of base log property, here this equation will become there is x minus 4 in the left hand side is equal to and the right hand side will become this is log of two with this 5.

[00:03:27]Here we will take this 4 to the right hand side.

[00:03:30]This gives him x is equal to 4 plus log of two with base 5.

[00:03:39]So, this is the final answer and a final value of x in terms of log.

[00:03:45]Here we will try to verify this value of x state A. This value of x verify this one beautiful algebraic problem equation or not?

[00:03:54]So, we'll verify this value here.

[00:03:57]To verify this one value here, we will write our question again. So, our question is 5 raised to power x divided by 25 is equal to 50.

[00:04:10]Before substitute the value of x here, first we will take this 25 to the right-hand side.

[00:04:15]This will become 5 raised to power x is equal to 25 * 50.

[00:04:25]I will substitute the value of x. So, this equation will become This will become 5 raised to power the value of x is 4 + log of 2 with base 5 is equal to 25 * 50.

[00:04:45]And in the left-hand side, we will use an exponential property.

[00:04:49]And we will use a raised to power m + n that can be written as a raised to power m * a raised to power n.

[00:05:00]So, using this exponential property in the left-hand side, this will become This will become 5 raised to power 4 * 5 raised to power log of 2 with base 5 is equal to 25 * 50.

[00:05:19]So, here we can write this 5 raised to power 4 as this is simply 5 squared * 5 squared.

[00:05:27]And here in this one term, we will use another log property.

[00:05:31]So, in this term, we will use this one log property. We can write a raised to power log of b with base a is equal to b.

[00:05:42]So, using this log property here, we will replace this whole number with only this number two.

[00:05:49]We come to is equal to 25 * 50.

[00:05:55]Here a 5 squared is simply 25.

[00:05:59]And this 5 squared is also 25. So, we come 25 times this is also 25. So, 25 * 2 8 this is 50.

[00:06:09]is equal to The right hand side is 25 times 50.

[00:06:16]Looking to both sides here, both sides are equal and same.

[00:06:20]So, this means that x is equal to 4 + log of 2 with base 5 is the exact and correct value of x in this interesting exponential and algebra math problem.