Exponents and Logarithms

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These are the solutions to the CSEC Additional Mathematics past questions on the topics: Exponents and Logarithms.
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These are the notable notes regarding factoring

Factoring Formulas

$ \underline{Difference\;\;of\;\;Two\;\;Squares} \\[3ex] (1.)\;\;x^2 - y^2 = (x + y)(x - y) \\[5ex] \underline{Difference\;\;of\;\;Two\;\;Cubes} \\[3ex] (2.)\;\; x^3 - y^3 = (x - y)(x^2 + xy + y^2) \\[5ex] \underline{Sum\;\;of\;\;Two\;\;Cubes} \\[3ex] (3.)\;\; x^3 + y^3 = (x + y)(x^2 - xy + y^2) \\[5ex] $ These are the laws of exponents and the laws of logarithms.

Laws of Exponents and Laws of Logarithms

Law 1: Exponents
(1.) Product Rule

$ p^c * p^d = p^{c + d} \\[4ex] p^{c + d} = p^c * p^d $

Law 1: Logarithms
(1.)

$ \log_p{c} + \log_p{d} = \log_p{cd} \\[4ex] \log_p{cd} = \log_p{c} + \log_p{d} $

Law 2: Exponents
(2.) Quotient Rule

$ p^c \div p^d = p^{c - d} \\[4ex] \dfrac{p^c}{p^d} = p^{c - d} \\[4ex] p^{c - d} = p^c \div p^d \\[4ex] p^{c - d} = \dfrac{p^c}{p^d} $

Law 2: Logarithms
(2.)

$ \log_p{c} - \log_p{d} = \log_p({c \div d}) \\[4ex] \log_p{c} - \log_p{d} = \log_p{\left(\dfrac{c}{d}\right)} \\[5ex] \log_p({c \div d}) = \log_p{c} - \log_p{d} \\[4ex] \log_p{\left(\dfrac{c}{d}\right)} = \log_p{c} - \log_p{d} $

Law 3: Exponents
(3.)

$ {any\: base}^0 = 1 \\[5ex] p^0 = 1 $

Law 3: Logarithms
(3.)

$ \log_{any\: base}{1} = 0 \\[5ex] \log_p{1} = 0 \\[4ex] \ln{1} = \log_e{1} = 0 $

Law 4: Exponents
(4.)

$ {any\: base}^1 = any\: base \\[5ex] p^1 = p $

Law 4: Logarithms
(4.)

$ \log_{any\: base}{any\: base} = 1 \\[5ex] \log_p{p} = 1 $

Law 5: Exponents
(5.) Expanded Power Rule

$ (p)^c = (p^1)^c = p^{1 * c} = p^c \\[5ex] \left(\dfrac{p}{q}\right)^c = \dfrac{p^c}{q^c} \\[7ex] (p^c)^d = p^{c * d} \\[5ex] p^{c * d} = (p^c)^d \\[7ex] \left(\dfrac{p^c}{q^d}\right)^e = \dfrac{p^{ce}}{q^{de}} \\[7ex] (pk)^d = p^d * k^d \\[5ex] p^d * k^d = (pk)^d \\[5ex] (p^c k^d)^m = p^{cm} * k^{dm} \\[5ex] p^{cm} * k^{dm} = (p^c)^m * (k^d)^m = (p^c k^d)^m \\[5ex] (p^c)^{\dfrac{d}{e}} = p^{\dfrac{cd}{e}} \\[7ex] p^{\dfrac{cd}{e}} = (p^c)^{\dfrac{d}{e}} $

Law 5: Logarithms
(5.)

$ \log_p{c^d} = d * \log_p{c} \\[4ex] d * \log_p{c} = \log_p{c^d} $

Law 6: Exponents
(6.) Rule of Negative Exponents

$ p^{-c} = \dfrac{1}{p^c} \\[5ex] \dfrac{1}{p^c} = p^{-c} \\[5ex] p^c = \dfrac{1}{p^{-c}} \\[5ex] \dfrac{1}{p^{-c}} = p^c \\[5ex] p^{-\dfrac{c}{d}} = \dfrac{1}{p^{\dfrac{c}{d}}} \\[7ex] \dfrac{1}{p^{\dfrac{c}{d}}} = p^{-\dfrac{c}{d}} \\[7ex] p^{\dfrac{c}{d}} = \dfrac{1}{p^{-\dfrac{c}{d}}} \\[7ex] \dfrac{1}{p^{-\dfrac{c}{d}}} = p^{\dfrac{c}{d}} $

Law 6: Logarithms
(6.)
Change of Base of Log

$ \log_p{d} = \dfrac{\log_c{d}}{\log_c{p}} \\[4ex] \dfrac{\log_c{d}}{\log_c{p}} = \log_p{d} \\[4ex] \log_p{d} * \log_c{p} = \log_c{d} \\[4ex] \log_c{d} = \log_p{d} * \log_c{p} $

Law 7: Exponents
(7.) Rule of Fractional Exponents

$ p^{\dfrac{1}{c}} = \sqrt[c]{p} \\[5ex] p^{\dfrac{c}{d}} = \sqrt[d]{p^c} \\[5ex] p^{\dfrac{c}{d}} = (\sqrt[d]{p})^c \\[5ex] \sqrt[d]{p^c} = p^{\dfrac{c}{d}} \\[5ex] (\sqrt[d]{p})^c = p^{\dfrac{c}{d}} $

Law 7: Logarithms
(7.)

$ p^{\log_p{c}} = c \\[5ex] c = p^{\log_p{c}} \\[5ex] p^{d\log_p{c}} = p^{\log_p{c^d}} = c^d \\[5ex] c^d = p^{\log_p{c^d}} = p^{d\log_p{c}} \\[5ex] e^{\ln{c}} = c \\[5ex] c = e^{\ln{c}} \\[5ex] e^{d\ln{c}} = e^{\ln{c^d}} = c^d \\[5ex] c^d = e^{\ln{c^d}} = e^{d\ln{c}} $

Formula Sheet: List of Formulae

(1.) (a.) Write the expression $2\log_3{x} + 2 - \log_3{y}$ as a single term.

(b.) (i.) By using logarithms, express the relationship $V = 7 \times 5^t$ in linear form.

(ii.) Hence, state the value of the gradient of the line which represents the relationship in (b.)(i.)

$ (a.) \\[3ex] 2\log_3{x} + 2 - \log_3{y} \\[4ex] 2\log_3{x} = \log_3{x^2} ...Law\;5...Log \\[5ex] 2 = 2\log_3{3} = \log_3{3^2} ...Laws\;4\;\;and\;\;5...Log \\[5ex] \implies \\[3ex] \log_3{x^2} + \log_3{3^2} - \log_3{y} \\[5ex] \log_3{x^2} + \log_3{9} - \log_3{y} \\[5ex] \log_3\left({\dfrac{9x^2}{y}}\right)...Laws\;1\;\;and\;\;2...Log \\[5ex] (b.)(i.) \\[3ex] V = 7 * 5^t \\[4ex] \text{Introduce Logarithm to both sides} \\[3ex] \log V = \log (7 * 5^t) \\[4ex] \log V = \log 7 + \log 5^t ...Law\;1...Log \\[4ex] \log V = \log 7 + t\log 5 ...Law\;5...Log \\[3ex] \log V = t\log 5 + \log 7 \\[3ex] Compare\;\;to\;\; y = mx + b ...\text{Slope-Intercept Form of a Linear Function} \\[3ex] $ (ii.) The gradient of the line which represents the relationship between $\log V$ and t is $\log 5$

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Exponents and Logarithms

Welcome to Our Site

Factoring Formulas

Laws of Exponents and Laws of Logarithms

Law 1: Exponents

Law 1: Logarithms

Law 2: Exponents

Law 2: Logarithms

Law 3: Exponents

Law 3: Logarithms

Law 4: Exponents

Law 4: Logarithms

Law 5: Exponents

Law 5: Logarithms

Law 6: Exponents

Law 6: Logarithms

Law 7: Exponents

Law 7: Logarithms

Formula Sheet: List of Formulae