Without using a calculator, find the exact value of log4 [log 3 (...

Question

Without using a calculator, find the exact value of log4 [log 3 (log₂ 8)].

Accepted Answer

Hello. Today we're gonna be simplifying the given logarithm. So the given logarithm is the logarithm of base nine of the argument logarithm of base two of the logarithm of base seven of 49. So this looks like a jumble of logarithms. But in order to start this, let's go ahead and start by simplifying the innermost logarithm. So the innermost logarithm is the logarithm of base seven of 49. Now 49 can be rewritten as seven squared. So we can rewrite the insight argument as seven squared or in other words, we have the logarithm of base seven of seven squared. Now, we don't want to have this exponent inside the argument of the logarithm. So using the exponential rule, if we have the logarithm of A to the power of B, we can go ahead and bring this exponent of B in front of the logarithm as a product. Or in other words, the logarithm of A to the power of B can be rewritten as B times the logarithm of A. So we can go ahead and bring this exponent of two in front of the logarithm. And what we are left with is two times the logarithm of seven of seven. Now using the property, which states that if you have the logarithm whose base matches his argument, you can go ahead and rewrite that logarithm as one. And here we have a base of seven and argument of seven. So the base and the argument match. So this logarithm can be rewritten as one. And what we are left with is two times one, which is going to give us two. So this innermost logarithm can be rewritten as two. And what we are left with is the logarithm of base nine times the logarithm of two of base two of two. Now using this logarithmic property, again, we have a base of two and an argument of two. So this logarithm can once again be rewritten as one. And what we are left with is the logarithm of nine of one. Now, there is no other way to rewrite the insight argument. So we know that this has to equal to some certain value. So let's go ahead and set this logarithm equal to X. And now we want to go ahead and sell for X. Well, in order to get rid of the logarithm, we can go ahead and take our base nine and set it to an exponent of X or in other words, we can rewrite this equation as nine to the power of X equal to one. Now, there is only one value of X that makes this statement true. If X is equal to zero, then what we are left with is nine to the power of zero is equal to one. And any number to the power of zero can be rewritten as one. So this statement gets rewritten as one is equal to one, which is true. So X has to equal to zero. And that means that this logarithm which is the logarithm of nine base nine of one is equal to zero. And this is gonna be the simplified version of our given logarithm. And that means that the answer to this problem is going to be B so I hope this video helps you in understanding how to simplify this given logarithm. And I'll go ahead and see you all in the next video.

Without using a calculator, find the exact value of log4 [log 3 (log₂ 8)].

Key Concepts

Logarithmic Functions

Change of Base Formula

Properties of Logarithms

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