5.3 Equations Involving Indices (Example 3)

Posted on April 21, 2020 by user

Example 3 (Index Equation - Equal Base)
Solve each of the following.
(a)

$27 (81^{3 x}) = 1$
(b)

$\sqrt{81^{n + 2}} = \frac{1}{3^{n} 27^{n - 1}}$
(c)

$8^{x - 1} = \frac{4}{\sqrt{2^{x + 3}}}$

5.3 Equations Involving Indices

Posted on April 21, 2020 by user

METHOD:

Comparison of indices or base

If the base are the same , when $a^{x} = a^{y}$ , then x = y
If the index are the same , when $a^{x} = b^{x}$ , then a = b

Using common logarithm (If base and index are NOT the same)

$\begin{array}{l} a^{x} = b \Rightarrow \lg a^{x} = \lg b \\ x = \frac{\lg b}{\lg a} \end{array}$

Example 1 (Index Equation - Equal base)
Solve each of the following.
(a)

$16^{x} = 8$
(b)

$9^{x} {.3}^{x - 1} = 81$
(c)

$5^{n + 1} = \frac{1}{125^{n - 1}}$

Example 2 (Solving index equation simultaneously)
Solve the following simultaneous equations.

$2^{x} {.4}^{2 y} = 8$

$\frac{3^{x}}{9^{y}} = \frac{1}{27}$

5.2b Change of Base of Logarithms (Example 5)

Posted on April 21, 2020 by user

Example 5:

Given that log₂ 3 = 1.585 and log₂ 5 = 2.322, evaluate the following.

log₈ 15
log₅ 0.6
log₁₅ 30
log₁₆ 45

Solution:

5.2b Change of Base of Logarithms (Example 3 & 4)

Posted on April 21, 2020 by user

Example 3
Given that

$\log_{p} 3 = h$ and

$\log_{p} 5 = k$ , express the following in term of h and or k.
(a)

$\log_{p} \frac{5}{3}$

(b)

$\log_{15} 75$

Example 4
Given that

$\log_{3} x = b$ , express

$\log_{x} 9 x$ in term of b.

5.2b Change of Base of Logarithms

Posted on April 21, 2020 by user

Change of Base of Logarithms

$\log_{a} b = \frac{\log_{c} b}{\log_{c} a} and \log_{a} b = \frac{1}{\log_{b} a}$

Example 1:
Find the value of the following:
a.

$\log_{25} 100$
b.

$\log_{3} 0.45$

Answer:

$\begin{array}{l} (a) \log_{25} 100 = \frac{\log_{10} 100}{\log_{10} 25} = \frac{\log_{10} 10^{2}}{\log_{10} 25} = \frac{2}{1.3979} = 1.431 \\ (b) \log_{3} 0.45 = \frac{\log_{10} 0.45}{\log_{10} 3} = \frac{- 0.3468}{0.4771} = - 0.727 \end{array}$

Example 2
Find the value of the following.
(a)

$\log_{4} 8$
(b)

$\log_{125} 5$
(c)

$\log_{81} 27$
(d)

$\log_{16} 64$

5.2a Laws of Logarithms (Example 3)

Posted on April 21, 2020 by user

Example 3

Given that

$\log_{7} 4 = 0.712$ and

$\log_{7} 5 = 0.827$ , evaluate the following.
(a)

$\log_{7} 20$
(b)

$\log_{7} 1 \frac{1}{4}$
(c)

$\log_{7} 0.8$
(d)

$\log_{7} 28$
(e)

$\log_{7} 140$
(f)

$\log_{7} 100$
(g)

$\log_{7} 0.25$
(h)

$\log_{7} \frac{35}{64}$

5.2a Laws of Logarithms (Example 2)

Posted on April 21, 2020 by user

Example 2

Find the value of the following.
(a)

$\log_{2} 7 + \log_{2} 12 - \log_{2} 21$

(b)

$3 \log_{10} 5 + 2 \log_{10} 2 - \log_{10} 5$

(c)

$2 \log_{10} 3 - \log_{10} 3 + \log_{10} 3 \frac{1}{3}$

(d)

$\log_{3} 3 p + \log_{3} 3 q - \log_{3} p q$

5.2a Laws of Logarithms (Example 1)

Posted on April 21, 2020 by user

Example 1

Express the following in term of

$\log_{a} x$ and

$\log_{a} y$ .
(a)

$\log_{a} 3 x$
(b)

$\log_{a} \frac{x}{5}$
(c)

$\log_{a} y^{5}$
(d)

$\log_{a} x y^{3}$
(e)

$\log_{a} \frac{x^{2}}{\sqrt{y}}$
(f)

$\log_{a} \sqrt{\frac{y}{a^{2} x^{3}}}$

Law of Logarithms

5.2a Laws of Logarithms

Posted on April 21, 2020 by user

5.2a Laws of Logarithms

$\begin{array}{l} Law 1: \\ {log}_{a} x y = {log}_{a} x + {log}_{a} y \\ Example: \\ \log_{5} 25 x = {log}_{5} 25 + {log}_{5} x \\ Beware!! \\ {log}_{a} x + {log}_{a} y \neq {log}_{a} (x + y) \end{array}$

$\begin{array}{l} Law 2: \\ {log}_{a} (\frac{x}{y}) = {log}_{a} x - {log}_{a} y \\ Example: \\ \log_{5} \frac{x}{25} = {log}_{5} x - {log}_{5} 25 \\ Beware!! \\ {log}_{a} \frac{x}{y} \neq \frac{{log}_{a} x}{{log}_{a} y} \end{array}$

$\begin{array}{l} Law 3: \\ {log}_{a} x^{m} = m {log}_{a} x \\ Example: \\ \log_{5} y^{5} = 5 {log}_{5} y \\ Beware!! \\ {({log}_{a} x)}^{2} \neq 2 {log}_{a} x \end{array}$

5.2 Logarithms

Posted on April 21, 2020 by user

5.2 Logarithms

$N = a^{x} \Leftrightarrow \log_{a} N = x$

$\log_{a} N = x$ is called the logarithmic form and

$N = a^{x}$ is the index or exponential form.

Note:

The logarithm of a negative number is not defined.
log in the calculator denotes $\log_{10}$ or common logarithm.
$\log_{10}$ may be written as lg.
If the base is other than 10, the base should be specified, e.g. $\log_{3} 81$