Solve for x in this nice exponential equation | Can you solve this? | Math Olympiad Mathematics

[00:00:00]In this video we want to solve for x given 10^ x + 10^ x + 10^ x / 5 raised to power x + 5^ x is = 100 to solve for x in this problem.

[00:00:30]Let us first factoriize this numerator.

[00:00:35]We have 10 raised to power x being added to itself 1 + 1 + 1 times and then we factoriize the denominator as well to give us 5 rais^ x into 1 + 1 cuz it's added to itself twice and then to give us 100.

[00:00:59]From here we have 10 raised to power x * 3 / 5 raised to power x * 2 is = 100 I can separate this like 10^ x / 5 raised to power x * 3 / 2 is = 100.

[00:01:32]By law of indices, given a raised to power m / b raised to power m, we can also express this as a / b raised to power m since they have the same powers.

[00:01:48]Then this becomes 10 / 5.

[00:01:54]Now both raised to power x then * 3 / 2 to give us 100.

[00:02:04]We now see that 10 is / 5 here to give us 2 and then we have 2 raised to power x * 3 / 2 is = 100.

[00:02:17]I'm going to divide both sides of this equation by 3 /2.

[00:02:24]That's 100 / 3 / 2. This takes care of this. And then we have 2 raised to power x is = 100 / 3 / 2.

[00:02:43]Then 2^ x is = 100 * 2 / 3. We change this to product and then we reciprocate the fraction. So that we now have 2^ x is equal to 100 * 2 will give us 200 then / 3 which is an exponential equation and we can take the logarithm of both sides. So that log 2 raised to power x is equal to log 200 / 3.

[00:03:27]We're going to apply two different kind of laws here. The first law of logarithm I'll apply here deals with the left hand side because the left hand side is of the form log p raised to power a and by love logarithm this will give us a * log p. So that this becomes x * log 2 is = log 200 divided by 3.

[00:04:01]Now the second law applies to the right hand side. Given log a / m by law of logarithm this will give us log a minus log m.

[00:04:18]Therefore x log 2 will now be equal to log 200 minus log 3.

[00:04:33]We can divide both sides by log two.

[00:04:46]So this divides this and then x becomes log 200 / log 2 - log 3 / log 2. Let us break down this 200 200 / 2 will give us 100 / 2 will give us 50.

[00:05:25]50 / 2 will give us 25.

[00:05:30]We know 25 is 5 squar. So 200 is equivalent to 2 * 2 * 2 is 8. So we have 2^ 3 * 25 is 5^ 2.

[00:05:45]Therefore x is now equal to log. So instead of writing log 200 I will write log 2^ 3 * 5^ 2 / log 2us log 3 / log 2.

[00:06:12]Then by love logarithm when given log a * b we can separate this into log a plus log b.

[00:06:26]So for this numerator here we're going to have log 2 raised to power 3 plus log 5 to power 2 / by log 2 then - log 3 / log 2. If we then divide or separate this division, we're going to have x is = to log 2^ 3 / log 2 + log 5 power 2 / log 2us log 3 / log 2 We then have 3 * log 2 / log 2 + 2 * log 5 / log 2 - log 3 / log 2 log two here cancels log two here. So x becomes 3 plus let's write this as 2 * log 5 / log 2 thenus log 3 / log 2.

[00:08:17]And now if we look at this expression as well as this expression we see they are both of the form log a / log p by law of logarithm this will give us log a base p. So this becomes log 5 is 2 and this becomes log 3 is 2. Then x becomes 3 + 2 * log 5 base 2 - log 3 base 2.

[00:08:56]This will now be our final answer to this problem. Before we conclude our final answer, let us do a quick check by substituting this answer back into our given problem. We are given 10^ x + 10^ x + 10^ x / 5 raised to power x + 5^ x to give us 100.

[00:09:30]Before we go ahead and substitute this gave us 10^ x * 3 then / 5^ x * 2 to give us 100.

[00:09:45]We'll make our substitution from here.

[00:09:47]Remember we got 3 + 2 log 5 is 2 - log 3.

[00:09:55]Therefore this will now be 10 raised to power x is 3 + 2 log 5 base 2 - log 3 base 2 all of that time 3 then divided by 5 raised to power 3 + 2 log 5 b 2 - log 3 base 2 all of that time 2 to give us 100.

[00:10:41]We need to separate these powers to be able to proceed from here. We use this law of indices.

[00:10:50]A raised ^ m + n will give us a raised power m * a raised to power n. So this then becomes 10^ 3 * 10 raised to power 2 log 5 B 2* 10 power log 3 B 2 then * 3 / 5 raised to power 3 * 5 ^ 2 log 5 B 2 * 5^ log 3 B 2 * 2 to give us 100.

[00:11:50]This two with negative powers will swap places because they have negative powers.

[00:11:57]Then we know that 10 to^ 3 is 1,000.

[00:12:02]So 1,00 time 10 raised to power. We can bring this two here to give us 5 to power 2. So this goes away. So this is 10 to power log 5 to power two is 25 is two.

[00:12:22]Then times since we said this was swap places then this becomes 5 raised to power this will be log 3 base 2 then * 3 divided by 5 to power 3 here is 125* 5 ra to power this two here will come here. So this goes away.

[00:12:56]Then we have log 5^ 2 which is 25.

[00:13:00]Then this two then times this will come down to give us 10 raised to power log 3 then 2.

[00:13:16]All of this to give us 100.

[00:13:22]25 in 125 here will give us five.

[00:13:26]Now 25 in 1,000.

[00:13:30]25 in 100 will give us 4. So 25 in 1,000 will give us 40. Then five here is 1. 5 in 40 is 8.

[00:13:41]Now 2 here is 1. 2 and 8 will give us 4.

[00:13:48]Then we have 4 * 3 here which will give us 12. So we have 12 * Now we need to handle here with care. We can split this into 2 raised to power log 25 base 2 * 5 ra to power log 25 base 2.

[00:14:16]Then time 5 to power log 3 this 2 / this is 5 raised to power log 25 base 2 and now we split this as well since 10 is 5 * 2 so this will be 2 raised to power log 3 B 2 * 5 raised to power log 3 B 2 then all of that to give us 100.

[00:15:00]So this is here we split this into this.

[00:15:04]So everything is intact. Then we see that this here cancels this and then this here cancels this and we are left with 12 * 2 raised to power log 25 base 2 / 2 raised to power log 3 base 2 to give us 100.

[00:15:40]Let us consider these two expressions involving logarithm. They are both of the form k raised to power log m k. By law of logarithm this will give us m. So here this will just give us 25 and this will give us 3. Then we have 12 * 25.

[00:16:04]then divided by three to give us 100.

[00:16:11]Three here is one. 3 in 12 is 4.

[00:16:16]That leaves us with 4 * 25 to give us 100.

[00:16:24]4 * 95 is 100.

[00:16:27]So we have 100 is equal to 100. And because the left hand side balances the right hand side, it confirms that our solution for x, which is 3 + 2 log 5 b 2 - log 3 b 2 is perfectly correct.

[00:16:46]Thanks for watching. Please like and share and also remember to subscribe to my channel if you have not done so already and I'll see you in my next video.

[00:16:56]Bye.