# Video: Solving Base e Exponential Equations

Given that π^(3π₯) β 2 = 1, find the value of π₯. If necessary, give your answer to 3 decimal places.

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### Video Transcript

Given that π to the power of three π₯ minus two equals one, find the value of π₯. If necessary, give your answer to three decimal places.

Weβve got an equation in terms of exponents of π₯. Then, weβre looking to find the value of π₯ to solve for π₯. Weβre going to begin by solving this as we would any other equation. Weβre going to begin by adding two to both sides. And we find that π to the power of three π₯ is equal to three. Now, we know that the inverse of the exponential function is the natural logarithmic function. So, weβre going to take the natural log of both sides. When we do, we find that the natural log of π to the power of three π₯ is equal to the natural log of three.

And then, we use one of the laws of logs. That is, log of any base of π to the power of π is equal to π times the log of π. So, we can rewrite the left-hand side as three π₯ times the natural log of π. But of course, the natural log of π is simply one. Now, we know that three π₯ is equal to the natural log of three, we can solve by dividing through by three. And π₯ is equal to the natural log of the three divided by three.

Typing that into our calculator, we find that π₯ is equal to 0.36620. Now, of course, weβre rounding that to three decimal places. Two is less than five, so we round down. And so, given that π to the power of three π₯ minus two equals one, we can say that the value of π₯ correct to three decimal places is 0.366.