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Questions and Answers
Which of the following expressions is equivalent to $(4 - 3)(6 + 2)$?
Which of the following expressions is equivalent to $(4 - 3)(6 + 2)$?
What is the result of the expression $5(3 - 2)$?
What is the result of the expression $5(3 - 2)$?
If $p = 3$ and $q = -13$, what is the result of $p + q$?
If $p = 3$ and $q = -13$, what is the result of $p + q$?
What is the output of the expression $(x + 9)(x^2 - 3x + 2)$ when $x = 1$?
What is the output of the expression $(x + 9)(x^2 - 3x + 2)$ when $x = 1$?
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Which of the following results corresponds to $3^{2} + 2^{3} + 30 = ?$?
Which of the following results corresponds to $3^{2} + 2^{3} + 30 = ?$?
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What is the value of $x$ in the equation $(x - 2)(x + 2)(2x + 1) = 0$?
What is the value of $x$ in the equation $(x - 2)(x + 2)(2x + 1) = 0$?
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What is the degree of the polynomial $x^4 + ax^3 + 5x^2 + x + 3$?
What is the degree of the polynomial $x^4 + ax^3 + 5x^2 + x + 3$?
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Determine the value from the expression $(2a + b)^{2}$ when $a = 1$ and $b = 2$.
Determine the value from the expression $(2a + b)^{2}$ when $a = 1$ and $b = 2$.
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For the equation $3x - 4 = 0$, what is the value of $x$?
For the equation $3x - 4 = 0$, what is the value of $x$?
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Which identity will help in simplifying $ ext{log}_2(x - 2y) = 5$?
Which identity will help in simplifying $ ext{log}_2(x - 2y) = 5$?
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If $f(x) = x^4 - 3x^3 + 2x^2 + 5x - 1$, what is the value of $f(2)$?
If $f(x) = x^4 - 3x^3 + 2x^2 + 5x - 1$, what is the value of $f(2)$?
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When dividing polynomial $x^4 + ax^3 + 5x^2 + x + 3$ by $x^2 + 4$, what would you expect to obtain?
When dividing polynomial $x^4 + ax^3 + 5x^2 + x + 3$ by $x^2 + 4$, what would you expect to obtain?
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From the equation $P(x) = (x - c)Q(x) + R(x)$, what does $R(x)$ represent?
From the equation $P(x) = (x - c)Q(x) + R(x)$, what does $R(x)$ represent?
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Which expression represents the sum of logarithms: $2 ext{log}{36} 2 + ext{log}{36} 3$?
Which expression represents the sum of logarithms: $2 ext{log}{36} 2 + ext{log}{36} 3$?
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What is expected when evaluating $g(0)$ for $g(x) = 4x^5 - 3x^3 + 2x + 1$?
What is expected when evaluating $g(0)$ for $g(x) = 4x^5 - 3x^3 + 2x + 1$?
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Which value for $A$ can be determined from the equation $x^4 + x^2 + x + 1 hickapprox (x^2 + A)(x^2 - 1) + Bx + C$?
Which value for $A$ can be determined from the equation $x^4 + x^2 + x + 1 hickapprox (x^2 + A)(x^2 - 1) + Bx + C$?
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What is the correct expression for the decomposition of $(x^2 + 3x)$?
What is the correct expression for the decomposition of $(x^2 + 3x)$?
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Which of the following represents a valid operation involving quotients?
Which of the following represents a valid operation involving quotients?
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Which option corresponds to a valid expression for factoring the polynomial $x^2 - x + 1$?
Which option corresponds to a valid expression for factoring the polynomial $x^2 - x + 1$?
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What is the result of $a^m \cdot a^n$?
What is the result of $a^m \cdot a^n$?
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If $a^m = b^n$, what can be concluded about $\frac{a^m}{b^n}$?
If $a^m = b^n$, what can be concluded about $\frac{a^m}{b^n}$?
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What is the degree of the polynomial $2x^2 - 5 + 2$?
What is the degree of the polynomial $2x^2 - 5 + 2$?
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Which formula represents the logarithmic identity for $\log_a (xy)$?
Which formula represents the logarithmic identity for $\log_a (xy)$?
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Which expression correctly factors $x^3 + 3x$?
Which expression correctly factors $x^3 + 3x$?
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What is the value of $\log_a 1$?
What is the value of $\log_a 1$?
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Which expression is equivalent to $\log_a (x/y)$?
Which expression is equivalent to $\log_a (x/y)$?
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If $p$ and $q$ are rational numbers and $a, b$ are positive real numbers, which of the following is correct for $\left(\frac{a^p}{b^q}\right)^m$?
If $p$ and $q$ are rational numbers and $a, b$ are positive real numbers, which of the following is correct for $\left(\frac{a^p}{b^q}\right)^m$?
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For polynomials, which of the following terms represents the remainder in polynomial long division?
For polynomials, which of the following terms represents the remainder in polynomial long division?
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Which of the following is a correct statement about surds?
Which of the following is a correct statement about surds?
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What does the expression $a^m \div a^n$ simplify to?
What does the expression $a^m \div a^n$ simplify to?
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Which property of logarithms does the equation $\log_a a = 1$ illustrate?
Which property of logarithms does the equation $\log_a a = 1$ illustrate?
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What is the correct transformation of the equation $3 ext{log}_2 + 2 ext{log} 5 - ext{log} 20$?
What is the correct transformation of the equation $3 ext{log}_2 + 2 ext{log} 5 - ext{log} 20$?
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If $x = ext{log} y z$, $y = ext{log} z x$, and $z = ext{log} x y$, what can be concluded about the values of $x$, $y$, and $z$?
If $x = ext{log} y z$, $y = ext{log} z x$, and $z = ext{log} x y$, what can be concluded about the values of $x$, $y$, and $z$?
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What is the result of evaluating $( ext{log}_3 m)( ext{log}_m 81)$?
What is the result of evaluating $( ext{log}_3 m)( ext{log}_m 81)$?
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For the equation $3 ext{log}_c a - 2 ext{log}_c b + 1$, what can be inferred about the value it represents?
For the equation $3 ext{log}_c a - 2 ext{log}_c b + 1$, what can be inferred about the value it represents?
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If the equation $ ext{log}_{10}(x - y + 1) = 0$ is given, what does it imply about the variable $x$?
If the equation $ ext{log}_{10}(x - y + 1) = 0$ is given, what does it imply about the variable $x$?
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How can the expression $ ext{log}_{16}(xy)$ be rewritten?
How can the expression $ ext{log}_{16}(xy)$ be rewritten?
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Given the equation $6x + 1 = 18$, what is the value of $x$?
Given the equation $6x + 1 = 18$, what is the value of $x$?
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In the equation $ ext{log}_3(x + 2) = ext{log}_9(6x + 4)$, how is it simplified?
In the equation $ ext{log}_3(x + 2) = ext{log}_9(6x + 4)$, how is it simplified?
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Study Notes
Indices
-
Basic Rules:
- a^m * a^n = a^(m+n)
- (a^m)^n = a^(m*n) = (a^n)^m
- (ab)^m = a^m * b^m
- (a/b)^m = a^m / b^m
- (a^p * a^q)^m= a^(pm) * a^(qm)
- a^m / a^n = a^(m-n)
- (a^p / b^q)^m = a^(pm) / b^(qm)
- (a/b)^(-m) = (b/a)^m
-
Surds Theorem
- √m(b^n) = (√m b)^n = (√n b)^m = b^(n/m)
- √n(ab) = √n( a) * √n(b)
- √n(a^m) = a^(m/n)
- a^(n/m) = √m(a^n)
- √n(a/b) = √n(a) / √n(b)
-
Logarithms Theorem
- a^y = x, therefore y = log(a)x
- log(a) xy = log(a) x + log(a) y
- log(a) x^c = c * log(a) x
- log(a) (x / y) = log(a) x - log(a) y
- log(a) 1 = 0
- log(a) a = 1
- a^(log(a) x) = x
- log(b) N / log(a) N = log(b) a
- log(b) a = 1 / log(a) b
Polynomials
- P(x) = Q(x)D(x) + r(x)
- Q(x) = quotient
- D(x) = divisor
- r(x) = remainder
Examples
- Evaluate (log3m)(logm81)
- Simplify 81 as 3^4
- Apply log(a)a = 1 to simplify log(m) 81 = log(m) (3^4) = 4
- Answer = 4log(3)(m)
- Solve log(x) 8 = 1.5
- Write 1.5 as 3/2
- x^(3/2) = 8
- x = 8^(2/3)
- x = 4
- Simplify a. 3log2 + 2log5 - log20
- 3 log(2) + 2log(5) - log(20)
- Apply log(a) xy = log(a) x + log(a) y and log(a) (x / y) = log(a) x - log(a) y
- log(2^3) + log(5^2) - log(20)
- log (8 * 25 / 20) = log(10) = 1
- Show that log(bc) a = 5 log(c) a / 1 + log(c) b
- Apply log(a) a = 1 to simplify log(bc) a = log(c) a / log(c) bc
- Apply log(a) xy = log(a) x + log(a) y to simplify log(c) bc = log(c) b + log(c) c = 1+log(c) b
- log(bc)a = log(c)a / 1 + log(c)b
- Solve the simultaneous equations:
- log(2) (x-2y) = 5; log(2) x - log(4) y = 4
- Simplify log(4) y = log(2^2) y = 2log(2) y
- The equations are then: log(2)(x-2y) = 5; log(2)x - 2log(2)y = 4
- Solve as simultaneous linear equations in log(2)x and log(2)y
- log(2)x = 6; log(2)y = 1
- Calculate the solutions for x and y from log(2)x and log(2)y
- The solutions are x = 64, y = 2
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Description
Test your understanding of indices, surds, and logarithms with this quiz. Explore the basic rules and theorems related to these essential algebraic concepts. Challenge yourself with examples and evaluate logarithmic expressions.
