In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each ...

Question

In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C.

logb √(2/27)

Accepted Answer

Hello. Today we're going to rewrite the given expression in terms of M. And N. With the specified values for M. And N. So let's go ahead and take a look at the given expression. The expression given to us is the logo rhythm of base E. Of the square root of 8/49. And we are told that M. Is equal to the logarithms of base E of eight and N is equal to the logarithms of Basie of seven. So we're going to need to use the substitution after simplifying the given expression. So for the given expression we have the square root of 8/49 on the inside of the argument. So we can go ahead and rewrite the square root while applying the square root to both the numerator and denominator. So rewriting this is going to give us the logarithms of base E. Of the square root of eight Over the Square Root of 49. And the square root of 49 can be rewritten as seven. So what we have is the logarithms of Basie of the square root of 8/7. Now notice that we have a fraction on the inside argument of the logarithms and that's an indication that we should be using the division property for logarithms to rewrite the given expression and the division property states that if you have the logo rhythm of A over B. This can be rewritten as the logo rhythm of A minus the logarithms of B. Here are A is going to be the square root of eight and R. B. Is going to be seven. So we can rewrite are given expression as the logo rhythm of base E. Of the square root of eight minus the logarithms of Basie of seven. Now we can go ahead and rewrite the square root of eight as 8 to the power of a half. So let's just go ahead and do that. So we have the logarithms of Basie of eight to the power of a half. And now we want to go ahead and get rid of this exponent from the inside of the logarithms. We can do that using the exponential rule for logarithms and that states that if you have the logo rhythm of A. To the power of B. This can be rewritten as B times the logarithms of A. So what we can go ahead and do is to take this exponent of a half and move it to the front of the logarithms as a product. So what we are left with is one half times the logarithms of base E of eight minus the logarithms Of Base E. of seven. Now notice we have the logo rhythm of base E of eight, that is equal to M. And we have the logarithms of Basie of seven and that is equal to N. So we can replace this logo rhythm with em and this logarithm with N. And doing so is going to give us one half times M minus N. And this is going to be the complete simplification in terms of M and N. And that means that the answer to this problem is going to be a So I hope this video helps you in approaching this type of problem, and I'll go ahead and see you all in the next video.

In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C. logb √(2/27)

Key Concepts

Logarithmic Properties

Change of Base Formula

Square Roots and Exponents

Watch next