Another Log Question: how does '3^[log_3 (2^3)]' equal 2^3?

[Lime]

3^[log_3 (2^3)]

How do you get from that to this:

2^3

 

[stapel]

It's the basic definition of logarithms. The expression "log_b(a)" means "the power you would have to put on 'b' to get 'a'". For instance, log₅(25) = 2, because 5² = 25.

So "log₃(2³)" is "the power you would have to put on 3 in order to get 2³". Then they put this on 3. So what must you get?

Eliz.

 

[Lime]

So "log₃(2³)" is "the power you would have to put on 3 in order to get 2³". Then they put this on 3. So what must you get?

Why is it "in order to get 2^3?" Why 2^3?

 

[stapel]

Why is it "in order to get 2^3?" Why 2^3?

Um... because that's what the textbook author was in the mood for when he wrote the problem...?

Eliz.

 

[Lime]

There is no 2^3 in the problem. The original problem is

3^[3 log_3 2] = X

You have to find X.

 

[Denis]

Lime, LOOK at your original post: 3^[log_3 (2^3)]

I see 2^3 ; you don't? :shock:

 

[Lime]

I know that.

How is this:

3^[log_3 (2^3)]

Equal to this:

2^3 :?:

 

[stapel]

How is this: 3^[log_3 (2^3)]
Equal to this: 2^3

It's the basic definition of logarithms. The expression "log_b(a)" means "the power you would have to put on 'b' to get 'a'". For instance, log₅(25) = 2, because 5² = 25.

So "log₃(2³)" is "the power you would have to put on 3 in order to get 2³". Then they put this on 3. So what must you get?

The definition of "log" hasn't changed. The reasoning is, I'm afraid, still the same.

Eliz.

 

[Lime]

I see definitions but no reasoning.

 

[Mrspi]

I see definitions but no reasoning.

You have been given the definition of a log.....this problem is just an application of the definition.

log<SUB>b</SUB> a is the exponent you use on b to produce a.

If b<SUP>x</SUP> = a, then log<SUB>b</SUB> a = x, and b<SUP>log<SUB>b</SUB> a</SUP> HAS TO BE a.

You have 3 <SUP>log<SUB>3</SUB> 2<SUP>3</SUP></SUP>

What is log<SUB>3</SUB> 2<SUP>3</SUP>? It is the exponent you use on 3 in order to get 2<SUP>3</SUP>. Now, what should you get if you actually use this as an exponent on 3? (That's what your problem is showing!)

Here's an example of a similar problem:

What is 7<SUP>log<SUB>7</SUB> 5</SUP>?

Since log<SUB>7</SUB> 5 is the exponent you use on 7 to get a result of 5, when you actually use this as an exponent on 7, what should you get? 5, of course.

I can't think of any other way to explain the reasoning....if you still don't understand what is going on here, you need to "revisit" the basic concept of logs.

 

[Lime]

The original problem was to solve X:

3^[log_3 (2^3)] = X

Today the lesson was on Cancelling Equations. One of the rules was:

a^[log_a (X)] = X if X > 0

Based on that rule, I can now answer my original question. But if I run into a problem like "Find the answer to a^[log_b (X)]", I don't know what I'll do.