Japanese | A Nice Exponents Problem | Maths Olympiad

[00:00:00]Hello, you're welcome to solve this math problem of 4 ^ n is equal to 240 to find the value of n from this equation.

[00:00:11]Solution.

[00:00:16]From this our problem, which is 4 ^ n is equal to 240, in the first step we'll apply log in both sides. So, it will be log of this 4 ^ n is equal to log of this 240.

[00:00:36]Then this power of n will move to the base. So, it will be n log of 4 is equal to log of 240.

[00:00:45][clears throat] Then into here, we'll divide by log of 4 in both sides. So, here over log of 4 and into this side over log of 4.

[00:01:00]So, this log of 4 cancel this log of 4, then it will be n is equal to log of 240 over log of 4.

[00:01:14]Then in the next step, it will be n is equal to log of 240 to our fours, it has how many fours? So, from 240 divided by 4, it is 60. Then divided by 4, it is 15.

[00:01:33]So, 240 it is 4 * 4, which is 4 squared, then times this 15 bracket.

[00:01:42]Then over this log of 4.

[00:01:48]Then in the next step here, it will be n is equal to from log of 4 squared times 15, this part here is in the form of the rule which is >> [snorts] >> log of a * b bracket which is equal to log of a + log of b.

[00:02:13]So, we'll apply this rule here into this form.

[00:02:18]Then into our problem, it will be log of 4 squared then plus log of 15.

[00:02:30]Then over this log of 4.

[00:02:36]Then in the next step, it will be n is equal to here log of 4 squared, this squared will move to the base. So, it will be 2 log of 4 then plus this here log of 15.

[00:02:52]Then divide by log of 4 here we divide by log of 4 and into here over log of 4.

[00:03:01]So, here it will be n is equal to log of 4 cancels log of 4. Then it will be 2 plus log of 15 over log of 4.

[00:03:16]Then in the next step, it will be n is equal to 2 plus log of 15. 15 15 into prime numbers, it is 3 * 5 bracket then over log of 4, it will be log of 4 into prime numbers, it is 2 squared. [snorts] Then here it will be n is equal to 2 plus log of 3 * 5 is same as log of 3 plus log of 5.

[00:03:52]Then over log of 2 squared. This square will move to the base, so it will be 2 log of 2.

[00:04:01]Then in the next step, here it will be n is equal to 2 plus here 2 log of 2 will be divided in this part and this part, so it will be this over this. So, log of 3 over 2 log of 2 plus this over this.

[00:04:21]Log of 5 over 2 log of 2.

[00:04:27]Then in the next step, here it will be n is equal to 2 plus into here in the denominator 2 is common.

[00:04:37]So, we can take 1 over 2 outside the bracket. Then it will be this over this, so log of 3 over log of 2 plus here 1 over 2. We take 1 over 2 outside the bracket, so it will be log of 5 over log of 2. So, log of 5 over log of 2. Bracket.

[00:05:02]Then into here it will be n is equal to 2 plus a half then bracket log of 3 over log of 2 is same as log of 3 base of 2 plus log of 5 over log of 2 is same as log of 5 base of 2 bracket.

[00:05:23]So, this here is the final answer for the value of n into this our problem.

[00:05:30]Or, we have another method. So, let's call this second method.

[00:05:39]From our problem, which is 4 power of n is equal to 2 power of 4.

[00:05:48]Now, into here into this 4 here into prime numbers, it is 2 squared. Then, bracket this power of n is equal to 240. Let's find the prime numbers of 240.

[00:06:07]Here, it is divisible by 2, which is 120.

[00:06:11]Then, divided by 2, it is 60.

[00:06:15]Then, divided by 2, it is 30.

[00:06:19]Then, divided by 2, it is 15.

[00:06:23]Then, divided by 3, it is 5.

[00:06:27]Then, divided by 5, it is 1.

[00:06:30]Then, from 240, it is 2 * 2 * 2 * 2, which is 2 power of 4 * 3 * 5. So, * 3 * 5.

[00:06:41]Then, here it will be 2 power of 2 * n power of 2n is equal to 2 power of 4 * 3 * 5.

[00:06:54]Then, here we'll apply log in both sides. So, it will be log of 2 power of 2n is equal to log of this here. 2 power of 4 * 3 * 5 bracket.

[00:07:12]Then, this power of 2n will move to the base. So, it will be 2n log of 2 is equal to Here, log of 2 power of 4, it will be log of 2 power of 4, then plus log of 3.

[00:07:29]Then, plus log of 5.

[00:07:34]Then, into here it will be we'll divide by log of 2 power of log of I mean, we'll divide by these two terms log of two in both sides so as to get n. So, it will be over two log of two and here over two log of two.

[00:07:56]So, this and this will cancel. This and this will cancel. Then, it will be n is equal to into here, power four will move to the base so it will be four log of two plus log of three plus log of five.

[00:08:16]Then, here divide by two log of two. So, here it will be over two log of two here over two log of two over two log of two.

[00:08:31]So, here it will be n is equal to this and this will cancel. Then, four divide by two it is two plus into here is same as one over two. So, one over two times log of three over log of two.

[00:08:48]So, log of three over log of two plus one over two times this over this log of five over log of two.

[00:09:05]Then, in the next step it will be n is equal to two plus here one over two we take outside the bracket. So, this divide by one over two it is this log of three over log of two plus this divide by half it is this here log of five over log of two bracket.

[00:09:35]Then, here it will be n is equal to 2 plus a half bracket log of 3 over log of 2 is same as log of 3 base of 2 plus this over this to be log of 5 base of 2.

[00:09:55]So, this is our final answer.

[00:09:58]So, you can choose which method is the simplest.

[00:10:02]For me, I think the second method is the simplest method.

[00:10:05]Now, let's check this our answer if it is correct. So, to check from our problem which is 4 power of n is equal to 240.

[00:10:19]So, into power of n, we substitute this our answer. So, it will be 4 power of n, we substitute power of 2 plus a half bracket log of 3 base of 2 plus log of 5 base of 2 bracket is it equal to this 240?

[00:10:43]Then in the next step, from this part here, this part here is in the form 4 power of 2 plus this power half log of 3 base of 2 plus log of 5 base of 2 bracket. This part is in the form of the rule which is a power of m plus n which is equal to a power of m times a power of n. So, we'll apply this rule here.

[00:11:15]Then in the next step, here we'll apply this rule into this form. So, it will be 4 power of 2. So, here 4 power of 2 times 4 power of a half bracket log of 3 base of two plus log of five base of two bracket then this here is it equal to this 240.

[00:11:45]Then in the next step here four square four square it is 16 times four four is same as two square then bracket this power here power of a half bracket log of three base of two plus log of five base of two bracket is it equal to this 240.

[00:12:15]Then into here this two will cancel this two so it will be 16 times two power of this here power of log of three base of two plus log of five base of two is it equal to 240.

[00:12:35]Then here into here it will be 16 times this part here this part is in this form.

[00:12:47]Then we change into this form so it will be two two power of log of three base of two times two power of log of five base of two is it equal to 240.

[00:13:03]Then into here it will be 16 times two power of log of three base of two this is same as three times two power of log of five base of two this is same as five is it equal to 240.

[00:13:18]Then here multiply 16 times three is 48 times five is it equal to 240.

[00:13:29]Now here, 5 8 * 5 is 40 goes 4.

[00:13:34]4 * 5 is 20. 20 + 4 is 24.

[00:13:39]Is it equal to 240?

[00:13:42]So this here is left side and left hand side with right hand side are equal. So it is true for the value of n is 2 + 1/2 log of 3 base of 2 + log of 5 base of 2 bracket.

[00:14:03]So this here is correct.

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