I can't make sense of this log property explanation?

[roger12]

Homework Statement

I decided to cram these two unrelated question into one post, because they are too small and I don't want to crowd the forum with my many little bitty questions.


1. log(base A) of B= 1/[log(base B) of A]

because: if log(base B) of A=C, then B^C=A and so B=A^1/C

Hence, log(base A) of B= 1/C= 1/[log(base B) of A]

2. plot the graph of:

x^2+x+1: x<1, x=1 for [-3, 4] with intervals of 0.5

It's a part of a bigger graph.

Homework Equations

The Attempt at a Solution

1. I don't see how introducing C as equal to a different base and then carrying out a bunch of algebraic manipulations prove anything. Am I missing something here? Thank You.

2. x,y pairs (-3, 7), (-2, 3), (-1,1) I chose the whole numbers for x, because they are more convenient to me. If you look at this part of the graph in my book it goes through different points. Can, you please tell me where I went wrong? Thanks.

 

[eumyang]

1. log(base A) of B= 1/[log(base B) of A]

because: if log(base B) of A=C, then B^C=A...

Remember the for logarithms,
log_n x = y iff n^y = x

So, using this, if
log_B A = C, then B^C = A.

... and so B=A^1/C

From
B^C = A,
raise both sides to the exponent of 1/C, so
(B^C)^1/C = (A)^1/C,
or B = A^1/C.

... Hence, log(base A) of B= 1/C= 1/[log(base B) of A]

Using that definition of logarithms I gave earlier, Since
B = A^1/C,
log_AB = 1/C.

Substitute log_B A = C into the fraction, so
log_AB = 1/(log_B A)

 

[Mark44]

Homework Statement

2. plot the graph of:

x^2+x+1: x<1, x=1 for [-3, 4] with intervals of 0.5

It's a part of a bigger graph.

The Attempt at a Solution

2. x,y pairs (-3, 7), (-2, 3), (-1,1) I chose the whole numbers for x, because they are more convenient to me. If you look at this part of the graph in my book it goes through different points. Can, you please tell me where I went wrong? Thanks.

Apparently you are graphing the equation y = x² + x + 1, although what you showed is not an equation.

What does this part (in red) mean?

x^2+x+1: x<1, x=1[/color] for [-3, 4] with intervals of 0.5

The points you show, (-3, 7), (-2, 3), (-1,1), are on the graph of y = x² + x + 1. Does the graph in your book come from this equation?

 

[roger12]

Using that definition of logarithms I gave earlier, Since
B = A^1/C,
log_AB = 1/C.

Substitute log_B A = C into the fraction, so
log_AB = 1/(log_B A)

Awesome! Why I forgot "If A=B^C, then C=log(base B)A" is beyond me.

 

[roger12]

Apparently you are graphing the equation y = x² + x + 1, although what you showed is not an equation.

What does this part (in red) mean?


The points you show, (-3, 7), (-2, 3), (-1,1), are on the graph of y = x² + x + 1. Does the graph in your book come from this equation?

The parts in red mean x is less than or equal to 1.

And it IS an equation :

y=x^2+x+1: x< or =1

and

y= 3-x : x>1

for [-3, 4]

y is defined differently for different values of x. But both expressions are the parts of the same equation. The part of the graph defined by y= 3-x : x>1 for [-3, 4] looks right, but the other part defined by y=x^2+x+1: x< or =1 for [-3, 4] doesn't sit on the points made up of ordered pairs above.