An exponential brutality | Find a complex solution

Added: 2026-05-13

[00:00:00]Hi, welcome back once again. Today, we're going to solve this interesting exponential equation, which is very easy when we talk about the real value x, right? Because we can see clearly that 7 to the power of x here is equal to 7 squared. And since we have the same base, this shows that x is just simply equal to two, right? But this is for the real value of x. But what makes this question more interesting is because we are asked to find all complex solutions. Oh my god.

[00:00:35]So, let's get started.

[00:00:38]Now, from the right-hand side, 49 is the same as 49 multiplied by one.

[00:00:44]Now, we know that one is equal to e raised to the power of 2 i pi with the help of Euler, right?

[00:00:56]Then, generally, this is going to be equal to e raised to the power of i pi, then with 2k's multiples, right? Here, k is an integer.

[00:01:10]So, if k is equal to one, this is equal to one. If k is equal to two, we get 2 raised to the power of 4 i pi. If k is equal to three, we get e raised to the power of 6 i pi, and so on.

[00:01:21]Therefore, the equation becomes 7 to the power of x is equal to 49 multiplied by one, right? Here, 7 to the power of x is equal to 49 multiplied by e raised to the power of 2 i pi k.

[00:01:41]Then, here we're going to do the following.

[00:01:44]From the left-hand side, we apply this property, a to the power of b is equal to e raised to the power of b times the ln of a.

[00:01:53]From the right-hand side, we apply this one, k is equal to e raised to the power of ln of K.

[00:02:00]Therefore, the left-hand side becomes e to the x ln of 7.

[00:02:05]This is equal to 49 becomes e raised to the power of ln of Remember, it is 7 squared.

[00:02:12]Then multiply by This one becomes e raised to the power of 2 i pi K.

[00:02:19]Now, we have the same base and we have product, so we're going to add the exponents. And before that, we need to make use of power rule of logarithm. So, this becomes 2 ln of 7. Then add [clears throat] them up to get e to the x * ln of 7 is equal to e raised to the power of 2 ln of 7 + 2 i pi K.

[00:02:45]And from here, we can see we have the same base. Let's equate the exponents.

[00:02:50]So, this implies from here that x ln of 7 is equal to 2 ln of 7 + 2 i pi K.

[00:03:02]Now, divide both sides by ln of 7 from here.

[00:03:08]So, these both get cancelled.

[00:03:11]Therefore, here we have x is equal to So, split the denominator. 2 ln of 7 / ln of 7 will give us just 2 + then 2 i pi K divided by ln of 7. So, here K is an integer. So, this right over here is all complex or This right over here are all complex solutions. Thank you for watching. If you enjoyed the video, please kindly subscribe to this channel.

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