ACT math logarithms and matrices

If $\displaystyle 10^x = 1000$, what is x?

x = 3 because $\displaystyle 1000 = 10^3$.

True or false: $\displaystyle log_{10}(100) = 2$.

False. $\displaystyle log_{10}(100) = 2$ is correct.

A cell phone's value depreciates. Current value is $300.

If $\displaystyle V = 300e^{-0.1t}$, what is t when $V = 150? 1. $\displaystyle 150 = 300e^{-0.1t}$ 2. $\displaystyle e^{-0.1t} = 0.5$ 3. $\displaystyle -0.1t = log(0.5)$ 4. $\displaystyle t = -10log(0.5) \\ t ≈ 6.93$.

Fill in the blank: $\displaystyle log_{2}(x) + log_{2}(y) = log_{2}(?)$

$\displaystyle log_{2}(xy)$.

A sound intensity of 90 dB is given by $\displaystyle I = 10^{-12}W/m^2$. What is the intensity?

Intensity: $\displaystyle I = 10^{(90/10)} = 10^{9}W/m²$.

If $\displaystyle log_{b}(x) = 3$, what is x in terms of b?

x = $\displaystyle b^{3}$.

True or false: $\displaystyle log_{5}(25) = 1$.

False. $\displaystyle log_{5}(25) = 2$ because $\displaystyle 25 = 5^2$.

In a computer algorithm, $\displaystyle T(n) = log_{2}(n)$ indicates what?

Efficiency decreases logarithmically as n increases.

A population grows exponentially: $\displaystyle P(t) = P_0 e^{rt}$. What is the growth rate if P doubles in 5 years?

$\displaystyle P(5) = 2P_0$ leads to $\displaystyle r = \frac{log(2)}{5} \\ r ≈ 0.1386$.

If a $\displaystyle 1000 investment grows to$2000, how long at 5% annual interest?

Use $\displaystyle A = P e^{rt}$. 1. $\displaystyle 2000 = 1000e^{0.05t}$ 2. $\displaystyle 2 = e^{0.05t}$ 3. $\displaystyle t = \frac{log(2)}{0.05} \\ t ≈ 13.86$ years.

What is the base of $\displaystyle log_{4}(16)$?

Base is 4 because $\displaystyle 16 = 4^2$.

In a rental scenario, the rent increases by 10% each year. What is the formula?

$\displaystyle R(t) = R_0(1.1)^t$.

A bacteria culture grows from 100 to 800 in 3 hours. What is the growth rate?

Use $\displaystyle N(t) = N_0 e^{rt}$: 1. $\displaystyle 800 = 100 e^{3r}$ 2. $\displaystyle 8 = e^{3r}$ 3. $\displaystyle r = \frac{log(8)}{3} \\ r ≈ 0.7937$.

If $\displaystyle log_{3}(x) + log_{3}(27) = 5$, what is x?

$\displaystyle log_{3}(x) + 3 = 5 \\ log_{3}(x) = 2 \\ x = 3^2 = 9$.

True or false: $\displaystyle log(10^5) = 5$.

True. $\displaystyle log(10^5) = 5$, as it's base 10.

If $\displaystyle log_{b}(4) = 2$, what does b equal?

b = $\displaystyle 16$ because $\displaystyle 4 = b^2$.

Matrix addition: What does it require?

Two matrices of the same dimensions. Example: Adding a 2x2 matrix to another 2x2.

True or False: Matrices can only be added if they are square.

False. They can be added if they have the same dimensions.

Fill in the blank: The result of multiplying a 2x3 matrix by a 3x2 matrix is a _____ matrix.

2x2 matrix.

What is the determinant used for?

To determine if a matrix is invertible. If the determinant is 0, it is not invertible.

A store sells 3 types of shirts: A, B, and C. Represent sales.

Use a matrix like: \[ \begin{bmatrix} 100 & 150 & 75 \end{bmatrix} \] for quantities.

Matrix multiplication: Can it be commutative?

No, multiplication of matrices is generally not commutative. Example: A*B ≠ B*A.

Calculate: If A = \[ \begin{bmatrix} 1 & 2 \end{bmatrix} \] and B = \[ \begin{bmatrix} 3 \end{bmatrix} \], find AB.

AB = \[ \begin{bmatrix} 3 & 6 \end{bmatrix} \].

What is a real-life application of matrices?

Budgeting: Use matrices to track expenses across different categories.

Fill in the blank: The identity matrix is a matrix that, when multiplied, leaves the other matrix _____

unchanged.

True or False: Every matrix has an inverse.

False. Only non-singular (det ≠ 0) matrices have inverses.

How do you solve a system of equations using matrices?

Use the inverse of the coefficient matrix: \( Ax = b \) leads to \( x = A^{-1}b \).

A company has 3 employees. Represent hours worked.

Use a matrix: \[ \begin{bmatrix} 40 & 35 & 45 \end{bmatrix} \] for hours.

What is the result of \[ \begin{bmatrix} 1 & 2 \end{bmatrix} \begin{bmatrix} 3 \end{bmatrix} \]?

Result is \[ \begin{bmatrix} 3 & 6 \end{bmatrix} \].

Define: A zero matrix.

A matrix where all elements are zero. Example: \[ \begin{bmatrix} 0 & 0 \end{bmatrix} \].

Compare: Row matrix vs Column matrix.

Row: 1xN matrix; Column: Nx1 matrix. Different shapes, not interchangeable.

Matrix operations: Why are dimensions important?

Dimensions dictate whether operations (addition, multiplication) can be performed.

If log10(100) = x, what is x?

x = 2. Because 10^2 = 100.

log(a * b) = ?

log(a) + log(b). Multiplying inside the log results in addition.

True or False: log(1) = 0.

True. Any log base equals zero when the input is 1.

If log2(x) = 5, what is x?

x = 32. Because 2^5 = 32.

Fill in the blank: log(a/b) = ...

log(a) - log(b). Dividing inside the log results in subtraction.

Evaluate log3(27).

log3(27) = 3. Because 3^3 = 27.

If log5(x) = 3, then x = ...

x = 125. Since 5^3 = 125.

log10(0.01) = ?

-2. Because 10^(-2) = 0.01.

What does log(a^b) equal?

b * log(a). Exponentiation inside the log results in multiplication.

True or False: log(a) - log(b) = log(a/b).

True. This is a property of logarithms.

If log4(16) = x, find x.

x = 2. Because 4^2 = 16.

log(10^x) = ?

x. The log base 10 of a power simplifies directly to the exponent.

What is log(1000) in base 10?

log(1000) = 3. As 10^3 = 1000.

If logb(y) = 4, what does y equal?

y = b^4. By the definition of logarithms.

Evaluate log(10) + log(100).

log(10) + log(100) = 3. Because 1 + 2 = 3.

If log2(x) = 1, what is x?

x = 2. Because 2^1 = 2.

What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns, used to represent data or perform calculations.

Multiply these matrices: [[1, 2], [3, 4]] and [[5, 6], [7, 8]].

Result: [[19, 22], [43, 50]] - Matrix multiplication involves dot products of rows and columns.

Find the determinant of this matrix: [[2, 3], [1, 4]].

Determinant = 2*4 - 3*1 = 8 - 3 = 5.

True or False: A matrix can represent a system of equations.

True - Each row can represent an equation, and each column can represent a variable.

Fill in the blank: The identity matrix for 2x2 is ___.

[[1, 0], [0, 1]] - It acts like 1 in matrix multiplication.

What happens when you multiply a matrix by its inverse?

You get the identity matrix. A*A^{-1} = I.

Compare: Matrix addition vs. multiplication.

Addition: element-wise. Multiplication: involves rows and columns.

If A = [[1, 2], [3, 4]] and B = [[5], [6]], find AB.

Result: [[17], [39]] - Multiply rows of A by column of B.

What is the trace of a matrix?

The trace is the sum of the diagonal elements in a square matrix.

Calculate the determinant of this matrix: [[3, 5], [2, 4]].

Determinant = 3*4 - 5*2 = 12 - 10 = 2.

True or False: A 3x2 matrix can be multiplied by a 2x3 matrix.

True - The inner dimensions match (2). Result will be a 3x3 matrix.

Fill in the blank: The zero matrix has all elements equal to ___.

0 - It behaves like 0 in matrix addition.

What is a row echelon form?

A form of a matrix where each leading entry of a row is to the right of the leading entry of the previous row.

Identify the matrix: [[1, 2], [0, 1]].

This is an upper triangular matrix - all elements below the diagonal are zero.

If A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], find A + B.

Result: [[1, 3], [4, 4]] - Add corresponding elements of A and B.

What does a non-zero determinant indicate?

It indicates that the matrix is invertible and the system of equations has a unique solution.