Given that 𝑒 to the power of three 𝑥 minus two equals one, find the value of 𝑥. If necessary, give your answer to three decimal places.
We’ve got an equation in terms of exponents of 𝑥. Then, we’re looking to find the value of 𝑥 to solve for 𝑥. We’re going to begin by solving this as we would any other equation. We’re going to begin by adding two to both sides. And we find that 𝑒 to the power of three 𝑥 is equal to three. Now, we know that the inverse of the exponential function is the natural logarithmic function. So, we’re going to take the natural log of both sides. When we do, we find that the natural log of 𝑒 to the power of three 𝑥 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 𝑎 to the power of 𝑏 is equal to 𝑏 times the log of 𝑎. So, we can rewrite the left-hand side as three 𝑥 times the natural log of 𝑒. But of course, the natural log of 𝑒 is simply one. Now, we know that three 𝑥 is equal to the natural log of three, we can solve by dividing through by three. And 𝑥 is equal to the natural log of the three divided by three.
Typing that into our calculator, we find that 𝑥 is equal to 0.36620. Now, of course, we’re rounding that to three decimal places. Two is less than five, so we round down. And so, given that 𝑒 to the power of three 𝑥 minus two equals one, we can say that the value of 𝑥 correct to three decimal places is 0.366.