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 8

Accepted Answer

Hello. Today we're going to rewrite the given expression in terms of M&N. So the given expression is the logarithms of Basie of 343 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 use the substitution for M. And N. To rewrite this expression in terms of M. And N. So 343 is equal 27 cubed. So we can rewrite the inside arguments of the logarithms as seven cubed. So what we have is the logo rhythm of base E. Of seven to the power of three. And using the exponential property of logarithms, which states that if you have the logo rhythm of A to the power of B. You can rewrite this as B times the logo rhythm of A. Or in other words you can breathe this exponent of three in front of the logarithms as a product. So utilizing this property is going to give us three times the logarithm of base E of seven. And notice that the logarithm of base E of seven is equal to N. So we can replace this entire logarithms with N. And what that leaves us with is three times N. Which is the complete simplification in terms of M. And N. And that means the answer to this problem is going to be C. So I hope this video helps you in understanding how to approach 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 8

Key Concepts

Logarithmic Properties

Change of Base Formula

Exponential Relationships

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