Japanese | A Nice Exponents Problem | Math Olympiad

[00:00:00]Hello, you are welcome to solve this math problem of 5^ a is equal to 500 to find the value of a from this equation.

[00:00:12]Solution from this problem here which is 5^ a is equal to 500.

[00:00:24]In the first step we'll apply log in both sides. So it will be log of 5^ a is equal to log of 500.

[00:00:37]Then this power of a will move to the base. So it will be a * log of 5 is equal to log of 500.

[00:00:50]Then we'll divide by log of five in both sides. So here over log of five and here over log of 5. So this log of five will cancel this log of five. Then it will be a is equal to log of 500 over log of 5.

[00:01:15]Then here it will be a is equal to from log of 500 500 is equal to 5 * 100. So we substitute this. So it will be log of 500. We substitute here 5 * 100 bracket then over this log of five.

[00:01:42]Then into here it will be a is equal to log of 5 * 100. This part here is in the form of log of a * bracket which is equal to log of a plus log of b. So we'll apply this rule here into this form. Then here it will be log of five. So here log of 5 + log of 100 then over this log of 5.

[00:02:23]Then in the next step it will be a is equal to we divide by log of five in this part and this part. So it will be this over this. So log of 5 over log of 5 plus this over this log of 100 over log of 5.

[00:02:47]Then into here it will be a is equal to log of 5 / log of 5 it is 1. So 1 + log of 100 over log of five.

[00:03:03]Then into here it will be a is equal to 1 + log of 100 is same as 10 * 10 which is 10² over log of 5.

[00:03:20]Then in the next step it will be a is = 1 + into here this square will move to the base. So it will be 2 log of 10 over log of 5.

[00:03:36]Then here it will be a is = 1 + 2. From here log of 10 10 is equal to to 5. 10 into prime number it is 5 * 2. So into log of 10 we substitute here [snorts] 5 * 2 bracket then over this log of 5.

[00:04:05]Then here it will be a is = 1 + 2 log of 5 * 2 is in this form here. Then we change into this form. So here it will be bracket because it is this. So it will be log of 5 + log of 2 bracket then over this log of 5.

[00:04:35]Then into here it will be a is equal to 1 + here we multiply by log of two inside the bracket. So this time this is 2 log of 5 plus this time this is 2 log of 2 then over this log of 5.

[00:04:59]Then into here it will be a is = 1 + we divide by log of five in here and here.

[00:05:05]So it will be this over this. So 2 log of 5 over log of 5 plus this over this 2 log of 2 over log of 5.

[00:05:23]Then here it will be a is equal to 1 + here log of 5 / log of 5 it is 1. So 1 * 2 it is 2 + it will be 2 log 2 * log of 2 over log of 5.

[00:05:43]Then here it will be a is equal to into 1 + 2 it is 3 + 2 log of 2 over log of 5 is same as log of 2 base of 5. So this is the final answer for the value of a into this our problem or we have another simplest method to solve this problem. So let's call this second method from our problem which is 5^ a is = 500 into here it will be 5^ a is = 500.

[00:06:33]Let's find the prime factors of 500. So from 500 here it is divisible by 5 it is 100. Then divide by 5 it is 20. Then divide by 5 it is 4. Then divide by 2 it is 2. Then divide by 2 it is 1. So from 500 it is 5 * 5 * 5 which is 5^ 3 * 2 * 2 it is 2^ of 2 it is 500 into exponent.

[00:07:12]Then here we'll apply log in both sides.

[00:07:16]So it will be log of this 5^ a is equal to log of this here which is 5^ 3 * 2^ 2 bracket.

[00:07:31]Then in the next step this power of a will move to the base. So it will be a log of 5 is equal to into here log of 5^ 3 * 2^ 2 this part here is in this form is in this form this form here then we change into this form so it will be log of 5^ 3 then plus log of 2^ 2 log of 2^ power of two.

[00:08:07]Then in the next step it will be a log of 5 is equal to this power of three will move to the base. So it will be 3 log of 5 plus power of two will move to the base. So it will be 2 log of 2. Then into here we divide by log of five in both sides. So here over log of five and here over log of five.

[00:08:39]Then in the next step here this log of five will cancel as log of five. Then it will be a is equal to we divide by log of five in this part and this part. So it will be this over this. So 3 log of 5 over log of 5 plus this over this 2 log of 2 over log of 5. Then here it will be a is equal to log of 5 / log of 5 is 1.

[00:09:12]So it will be 1 * 3 is 3 plus here it is 2 log of 2 over log of 5 is same as log of two base of five. So this is our final answer. So you can choose which method is the simplest for you. I think the second method is the simplest. Now let's check our answer if it is correct.

[00:09:39]So from our problem which is 5^ a is = 500 we'll check by using this our answer. So it will be 5^ a we substitute our answer which is 3 + 2 + 2 log of 2 base of 5 is it = 500?

[00:10:04]Then into here this part here this part is in the form of the rule which is a power of m + n which is equal to a^ of m * a^ n. So we'll apply this rule here into this form. Then it will be 5^ 3 time 5 power 5^ 2 log of 2 base of 5. Then is it equal to 500.

[00:10:40]Then from 5^ 3 5 * 5 * 5 is 125. Then times here it will be five. This two will move to the power. So it will be log of 2² base of 5 is it = to 500 then it will be 125 * 5^ of log of 2² it is 4 base of 5 is it = 500?

[00:11:18]Then here from this part here five power log of four base of five this part is in the form of a power of log of b base of a which is equal to b.

[00:11:32]So we'll apply this rule into this part then it will be 125 times here this here is same as four. So * 4 is it = to 500.

[00:11:48]Then from 4 * 25 it is 100. So here two zeros go with one. 4 * 1 is 4 plus that one it is 5 is equal to 500. So left side and right side are equal. Then it is true for the value of a it is this 3 + 2 log of 2 base of 5.

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