Things 3 8 4 Equals

• Things 3 8 4 Equals 1/3
• Things 3 8 4 Equals 6

3×6 − 8×4 = 18 − 32 = −14 What is it for? The determinant helps us find the inverse of a matrix, tells us things about the matrix that are useful in systems of linear equations, calculus and more. 3/8 + 3/4 = 1 1/8 How did we solve the problem above? When we add two fractions, such as 3/8 + 3/4, we make sure that the two denominators are the same and then we simply add the numerators. When we add two fractions, such as 3/8 + 3/4, we make sure that the two denominators are the same and then we simply add the numerators.

Logarithms: Simplifying with
'The Relationship' (page 2 of 3)

Sections: Introduction to logs, Simplifying log expressions, Common and natural logs

• Simplify log₂(8).

This log is equal to some number, which I'll call y. This naming gives me the equation log₂(8) = y. Then the Relationship says:

2 ^y = 8

That is, log₂(8), also known as y, is the power that, when put on 2, will turn 2 into 8. The power that does this is 3:

2³ = 8

Since 2 ^y = 8 = 2³, then it must be true that y = 3, and I get:

Advertisement

log₂(8) = 3

• Simplify log₅(25).

The Relationship says that, since log₅(25) = y, then 5 ^y = 25. This means that the given log log₅(25) is equal to the power y that, when put on 5, turns 5 into 25. The required power is 2, because 5² = 25. Then 5² = 5 ^y = 25, so:


log₅(25) = 2

• Simplify log₆₄(4). Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

The Relationship says that this log represents the power y that, when put on 64, turns it into 4. Remembering that 4³ is 64, and remembering that fractional exponents correspond to roots, this means that the cube root of 64 is 4, so 64^(1/3) = 4. Then:

log₆₄(4) = ¹/₃

This last example highlights the fact that, to be able to work intelligently with logs, you need to be pretty good with your exponents. So take the time to review them, if you're feeling a little shaky.

• Simplify log₆(6).

The Relationship says that, since log₆(6) = y, then 6 ^y = 6. But 6 = 6¹, so 6 ^y = 6¹, and y = 1. That is:

log₆(6) = 1

This is always true: log_b(b) = 1 for any base b, not just for b = 6.

• Simplify log₃(1).

The Relationship says that, since log₃(1) = y, then 3 ^y = 1. But 1 = 3⁰, so 3 ^y = 3⁰, and y = 0. That is:

log₃(1) = 0

This is always true: log_b(1) = 0 for any base b, not just for b = 3.

• Simplify log₄(–16).

The Relationship says that, since log₄(–16) = y, then 4 ^y = –16. But wait! What power y could possibly turn a positive4 into a negative16? This just isn't possible, so the answer is:

no solution

This is always true: log_b(a) is undefined for any negative argumenta, regardless of what the base is.

• Simplify log₂(0).

The Relationship says that, since log₂(0) = y, then 2 ^y = 0. But wait! What power y could possibly turn a 2 into a zero? This just isn't possible, so the answer is:

no solution

This is always true: log_b(0) is undefined for any base b, not just for b = 2.

• Simplify log_b(b³).

The Relationship says that 'log_b(b³) = y' means 'b ^y = b³'. Then clearly y = 3, so:

log_b(b³) = 3

This is always true: log_b(bⁿ) = nfor any base b.

Some students like to think of the above simplification as meaning that the b and the log-base-b 'cancel out'. This is not technically correct, but it can be a useful way of thinking of things. Just don't say it out loud in front of your instructor.

• Simplify 2^log₂(9).

Remember that a logarithm is just a power; it's a lumpy and long way of writing the power, but it's just a power, nonetheless. The expression 'log₂(9)' means 'the power which, when put on 2, turns 2 into 9.' And they've put that power onto 2, which means that the 2 has been turned into 9!

Looking at it another way, '2^log₂(9) = y' means 'log₂(y) = log₂(9)' (which is the equivalent logarithmic statement), so y = 9. But y = 2^log₂(9), so 2^log₂(9) = 9.

While the second way is technically correct, I find the first way to be more intuitive and understandable. Either way, though, I get an answer of:

2^log₂(9) = 9

This last example probably looks very complicated, and, in the technical explanation, it is. Look instead at the intuitive explanation (in the first paragraph). Some students even view the above problem as the 2 and the log-base-2 as 'cancelling out', which is not technically correct, but can be a useful way of remembering how this type of problem works.

Instabro 3 0 1 – browser for instagram. To synopsize, these are the things you should know from this lesson so far:

• The Relationship: 'log_b(x) = y' means the same thing as 'b ^y = x'.
• Logarithms are really exponents (powers); they're just written differently.
• log_b(b) = 1, for any base b, because b¹ = b.
• log_b(1) = 0, for any base b, because b⁰ = 1.
• log_b(a) is undefined if a is negative.
• log_b(0) is undefined for any base b.
• log_b(bⁿ) = n, for any base b.

<< PreviousTop |1 | 2 | 3| Return to IndexNext >> http://ogwvsb.xtgem.com/Blog/__xtblog_entry/19076350-wireframesketcher-6-2-14#xt_blog.

The determinant of a matrix is a special number that can be calculated from a square matrix.

Things 3 8 4 Equals 1/3

A Matrix is an array of numbers:

A Matrix
(This one has 2 Rows and 2 Columns)

The determinant of that matrix is (calculations are explained later):

What is it for?

The determinant helps us find the inverse of a matrix, tells us things about the matrix that are useful in systems of linear equations, calculus and more.

Symbol

The symbol for determinant is two vertical lines either side.

Example:

|A| means the determinant of the matrix A

(Exactly the same symbol as absolute value.)

Calculating the Determinant

First of all the matrix must be square (i.e. have the same number of rows as columns). Then it is just basic arithmetic. Here is how:

For a 2×2 Matrix

For a 2×2 matrix (2 rows and 2 columns):

The determinant is:

Qr code generator wifi. |A| = ad − bc
'The determinant of A equals a times d minus b times c'

Example:

For a 3×3 Matrix

For a 3×3 matrix (3 rows and 3 columns):

The determinant is:

|A| = a(ei − fh) − b(di − fg) + c(dh − eg)
'The determinant of A equals . etc'

It may look complicated, but there is a pattern:

To work out the determinant of a 3×3 matrix:

• Multiply a by the determinant of the 2×2 matrix that is not in a's row or column.
• Likewise for b, and for c
• Sum them up, but remember the minus in front of the b

Things 3 8 4 Equals 6

As a formula (remember the vertical bars || mean 'determinant of'):

'The determinant of A equals a times the determinant of . etc'

Example:

For 4×4 Matrices and Higher

The pattern continues for 4×4 matrices:

• plusa times the determinant of the matrix that is not in a's row or column,
• minus b times the determinant of the matrix that is not in b's row or column,
• plus c times the determinant of the matrix that is not in c's row or column,
• minus d times the determinant of the matrix that is not in d's row or column,

As a formula:

Notice the +−+− pattern (+a. −b. +c. −d.). This is important to remember.

The pattern continues for 5×5 matrices and higher. Usually best to use a Matrix Calculator for those!

Not The Only Way

This method of calculation is called the 'Laplace expansion' and I like it because the pattern is easy to remember. But there are other methods (just so you know).

Summary

For a 2×2 matrix the determinant is ad - bc
For a 3×3 matrix multiply a by the determinant of the 2×2 matrix that is not in a's row or column, likewise for b and c, but remember that b has a negative sign!
The pattern continues for larger matrices: multiply a by the determinant of the matrix that is not in a's row or column, continue like this across the whole row, but remember the + − + − pattern.