Properties of logarithms Assume logbx...

Question

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.

logb (√x) / (³√z)

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. Given that the log base A of P is equal to 0.42 and log base A of R is equal to 0.91, find the value of log base A of the square root of P divided by the fourth root of R. Awesome. So it appears for this particular problem, we're trying to figure out the value of this new log value that's given to us. So we're trying to figure out the value of log base A of the square root of P divided by the 4th root of R. So we're trying to figure out what the value of that specific log value is. Awesome. So with that in mind, let's read off our multiple choice answers to see what our final answer might be. A is 0.1075. B is -0.0175. C is 0.1750. And finally, D is -0.0715. OK. So first off, in order to help us solve this problem, let us define all of our known variables. So first off, we're told that log base A of P is equal to 0.42. Awesome, we're also told that log base A. Of Q Which they say the log base of Cuba, it's log base A of R. Is going to be equal to So log base A of R is equal to 0.91, and we're also told that log base A of Q, so we do have a Q, and the log base A of Q is equal to 0.67. Awesome. And our target variable, let's write this in blue, our target variable that we're ultimately trying to solve for is log base A of the square root of P divided by the fourth root of R. Fantastic. So, our next step is we need to recall and use the properties of logarithms to express the target variable. So this means that we need to recall a note that we could write that log base A. Of the square root of P divided by the 4th root of R. is equal to log base A of the square root of P minus the log base A of the 4th root of R. Fantastic. So Our next step now is we need to recall and apply the power rule of logarithms, so we need to take log base A of the square root of P is equal to log base A of P to the power of 12, which is equal to 1/2 multiplied by log base A of P. And we also need to recall and note that we could write that log base A of the 4th root of R is equal to log base A of R to the power of 1/4, which is equal to 1/4 multiplied by the log base A of R. Fantastic. So now, our next step is we need to substitute in our known variables into the equation, so we need to take or into our equations that we just found together, so we need to take one half multiplied by the log base A. So you take log base A. Of P is equal to and now when we plug in our non variables, we need take one half multiplied by 0.42, which is equal to 0.21, and we need you to take 14th multiplied by log base A of R, which is equal to 1/4 multiplied by 0.91, which is equal to 0.2275. Fantastic. So now, Our next step is we need to combine the results to help us find our target variable, which is the final answer that we're ultimately trying to solve for. So, meaning we need to subtract. 0.21. So let's write this in red. So we need to take 0.21, and then we need to subtract 0.2275. So when we do that, we will find that it's equal to, and we'll write this down below since we're running out of room, negative 0.0175, and that is it. That is our final answer. Hooray, we did it. So what we did, let's quickly recap here, is we took our results for our for our different values. So for once again we took Our known variables and we plugged them into our equations. So using our equations that we derived together, we plugged in our known variables and then we had to combine the results to find our target variables. So when we combined, and once again, this is stating that log base A of the square root of P divided by the 4th root of R is equal to. So they were essentially we're saying. And let's move this over ever so slightly. So we're saying that log base A of the square root of P divided by the 4th root of R, is going to be equal to 0.21 minus 0.2275, which will give us a grand total of negative 0.0175, and this corresponds to multiple choice letter B, which once again is -0.0175. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.

logb (√x) / (³√z)

Key Concepts

Properties of Logarithms

Change of Base Formula

Radicals and Exponents

Watch next