This Log Equation Looks Impossible Until You See this Trick.

Added: 2026-06-05

[00:00:00]Hello and welcome. In this math tutorial, our task is to find all the real values of X that satisfy this logarithmic equation.

[00:00:08]Now, to solve this problem, remember that log to base E of E is equal to 1.

[00:00:19]So, that means that we can write this one as log to base 5 since all the other terms are in base 5 of 5.

[00:00:31]Now, we can write this equation as log to base 5 of 3 to the power 2 X + 6 is equal to log to base 5 of 3 to the power X + log to base 5 of 5.

[00:00:54]And of course, you know that we can combine these two terms. Remember that log A + log B is equal to log A * B. That means that this right-hand side is exactly the same as log to base 5 of 3 to the power X multiplied by 5. And of course, on the left-hand side we still have log to base 5 of 3 to the power 2 X + 6.

[00:01:33]Now, look at this. Here, we have log to base 5 of this. And on the right-hand side you have log to base 5 of that.

[00:01:41]This simply means that 3 to the power 2 X + 6 is equal to 5 * 3 to the power X.

[00:01:52]Now, let us rearrange this equation. We have 3 to the power 2 x 5 * 3 to the power x + 6 is equal to 0.

[00:02:07]Now, let us simplify this equation with the substitution.

[00:02:11]Let 3 to the power x be equal to the letter D.

[00:02:17]Here, we have D squared, because of course, you know that we can also write this as 3 to the power x squared. From our laws of indices, to open this bracket, we simply multiply these two powers. And when we do that, we have 2 x, which is exactly what we have here. So, this is D squared, then 5 D + 6 is equal to 0.

[00:02:46]Now, we can solve this quadratic equation by factorization.

[00:02:50]Since the coefficient of D squared is 1, all we need to do is to find the factors of 6 that add up to -5.

[00:02:59]And of course, those are going to be -2 and -3. Because -2 * -3 is going to give us +6, while -2 -3 is going to give us -5.

[00:03:12]So, when we factorize this left-hand side, we have D -2 multiplied by D -3 is equal to 0.

[00:03:24]And of course, from here, it is easy to see that either D -2 is equal to 0.

[00:03:31]From where D is equal to 2.

[00:03:35]Or D -3 is equal to 0.

[00:03:40]From where D is equal to 3. So, we have two values of D, which means that 3 to the power x can either be equal to two or three to the power x is equal to three.

[00:03:59]Now, in this case, since [clears throat] we can't express two as an integer power of three, to solve this equation, we start by taking log of both sides of the equation. So, we have that log three to the power x is equal to log two. Now, let us apply the power rule of logarithms here. That is the rule that says that we can bring down the exponent x to the front of the equation to become a multiplier.

[00:04:27]When we do that, we have x times log three is equal to log two. And of course, the next thing we are going to do is to divide both sides of this equation by log three.

[00:04:43]This is going to take care of that.

[00:04:46]We have that x is equal to log two over log three. And of course, you know that according to the change of base formula, we can also write this as log to base three of two.

[00:05:08]Now, in the second case, of course, you know that we can express three as an integer power of three. This is three to the power one.

[00:05:16]Now that we have the same base on both sides of the equation, we simply equate the exponents. So, from here, we have that x is equal to one. So, the two values of x that satisfy this logarithmic equation are x equal to log to base three of two and x equal to one.

[00:05:38]Now, to check our answer, let us substitute these two values of x in turn into the original equation.

[00:05:45]When x is equal to log to base three of two, the equation becomes log to base five of three to the power two times log to base three of two plus six. Now, is this equal to log to base five of three to the power log to base three of two plus one?

[00:06:14]Now, here, let us apply the power rule.

[00:06:17]We are going to pick up the multiplier two to become the exponent of two. When we do that, on the left-hand side, we have log to base five of three to the power log to base three of two squared, then plus six.

[00:06:37]Now, let us look at this. Remember that a to the power log to base a of b is equal to b. So, that simply means that this is the same as log to base five of two. Because here, we have three to the power log to base three. So, this is log to base five of two plus one.

[00:07:04]Of course, you know that two squared is four. So, on the left-hand side, we have log to base five of three to the power log to base three of four, then plus six. Now, is this equal to log to base five of two plus one?

[00:07:24]Once more, you can see that here, we have three to the power log to base three. So, that means that this is the same as log to base five of four plus six. Now, is this equal to log to base five of two plus one.

[00:07:45]4 + 6 is 10. So, on the left-hand side we have log to base five of 10. But, remember that 10 is 5 * 2.

[00:07:57]Now, is this equal to log to base five of two plus one?

[00:08:12]And of course, you know that another way of writing this left-hand side is log to base five of five plus log to base five of two.

[00:08:24]Now, is this equal to log to base five of two plus one?

[00:08:30]Log to base five of five is one. So, this is 1 + log to base five of two, which is exactly equal to log to base five of two plus one.

[00:08:45]That means that when x is equal to log to base three of two, the left-hand side is equal to the right-hand side.

[00:08:54]Now, when x is equal to one, this equation becomes log to base five of 3 squared plus six.

[00:09:06]Now, is this equal to log to base five of three plus one?

[00:09:13]3 squared is nine. 9 + 6 is 15. So, on the left-hand side we have log to base five of 15.

[00:09:23]Is this equal to log to base five of three plus one?

[00:09:31]Now, of course, you know that 15 is 3 * 5. So, we can write this left-hand side as log to base 5 of 3 times 5. Now, is this equal to log to base 5 of 3 plus 1?

[00:09:45]Once more, according to the addition law, we can write this as log to base 5 of 3 plus log to base 5 of 5.

[00:09:58]Now, is this equal to log to base 5 of 3 plus 1?

[00:10:04]But of course, we know that log to base 5 of 5 is equal to 1. So, that means that on the left hand side, we have log to base 5 of 3 plus 1.

[00:10:16]And this is equal to log to base 5 of 3 plus 1. So, once more, when x is equal to 1, the left hand side is equal to the right hand side.

[00:10:30]And finally, we come to the end of this tutorial. I hope you learned something new. If you enjoy such contents, please subscribe to the channel. Leave us a like to support the channel. Thanks for watching. And you can see more tutorials here.