For a certain constant a1, ln a≈3.8067

Question

For a certain constant a>1, ln a≈3.8067 . Find approximate values of log₂ a and logₐ 2 using the fact that ln 2≈0.6931.

Accepted Answer

Hello there, today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. If a constant D is greater than one as a natural logarithm value of natural log of D is approximately equal to 5.2983 estimate the values of log base five of D and log base d of five using the fact that the natural log of five is approximately equal to 1.6094. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. We're trying to figure out the estimated values of our first answer which is log base five of D. And our second answer, we're also trying to estimate the value of log base D of five. Awesome. So now that we know that we're trying to estimate the values of two different log values. Let us read off our multiple choice answers to see what our final answer pair might be noting that our first answer will be the log base five of D is approximately equal to sum value. And our second answer will be log base D of five will be approximately equal to sum value. So A is 3.2921 and 0.3038. B is 3.2921 and 0.2038. C is 2.2921 and 0.3038. And finally, D is 2.2921 and 0.2038. Awesome. So first off, we need to estimate, so we're gonna be estimating log base five of D. So first off, we need to define the known variables. So we're told that the natural log of D is approximately equal to which we use our little curly equal signs, which natural log of D is approximately equal to 5.2983. And we're also told that the natural log of five is approximately equal to 1.6094. Fantastic. So now we need to recall and use the change of base formula for logarithms, meaning we have to take log base five D is equal to the natural log of D divided by the natural log of five. So at this point, we now need to substitute in our known variables into our equation, meaning we need to take log base five of D is equal to plugging in our known variables. 5.2983 divided by 1.6094 which will be approximately equal to when we round 212345 variables when we round 25 variables, which will be 3.29210. Fantastic. And apparently we're just rounding to 44 decimal places because as we could see in our multiple choice answers, they're all rounded to four decimal places. So we could just easily remove the zero. So our final answer is 3.2922921. So 3.2921 and that is our first answer, but we did it. Now, we can start solving for our second answer. So now we're gonna be estimating the log base D of five. So we're focusing now on log base D of five. Now that is our new value that we're trying to evaluate. Now. So we once again, we could define our known variables. So we know that log base D is 5.2983 and log five or the natural Laga five. So the natural Laga D as we know is our 5.2983. And we also know the natural Laga five is 1.6094. So once again, we need to recall and use our change of base formula for logarithms. So when we take log base D of five is equal to the natural log of five divided by the natural log of D. So now we just need to plug in our known variables and we need to plug in our known variables and just solve. So we need to take 1.6 094 divided by 5.2983. And we do that and we plug it into our calculator and we round to four decimal places. We will find that our final answer is equal to 0.3038 and that's it. We've officially solved for our, our final answers. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be multiple choice answer letter A which is log base five of D is approximately equal to 3.2921 and log base D of five is approximately equal to 0.3038. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

For a certain constant a>1, ln a≈3.8067 . Find approximate values of log₂ a and logₐ 2 using the fact that ln 2≈0.6931.

Key Concepts

Change of Base Formula

Natural Logarithm (ln)

Approximation of Logarithmic Values

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