Index Part2

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Definition of Logarithm

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

Note : It is VERY important to know how to change from index form to log form and vice versa

Value of a & N

Special notation

Examples

Exercise 1

Complete the following table

1 2^{3}=8\, \log_{2}8=3\,
2 \frac{1}{9}=3^{-2} \log_{3}\frac{1}{9}=-2\,
3 10^{2}=100\, \log_{10}100=2\, or \lg 100=2\,
4 10^{x}=y\, \log_{10}y=x\, or \lg y=x\,
5 e^{a}=b\, \log_{e}b=a\, or \ln b=a\,
6 e^{0}=1\, \log_{e}1=0\, or \ln 1=0\,
7 2^{5}=x\, \log_{2}x=5\,
8 x^{5}=2\, \log_{x}2=5\,
9 10^{y}=x\, \lg x=y\,
10 e^{x}=y\, \ln y=x\,

Laws of Logarithm

a) \log_{a}xy\, =\log_{a}x+\log_{a}y\,

b) \log_{a}\left(\frac{x}{y}\right)\, =\log_{a}x-\log_{a}y\,

c) \log_{a}x^{n}\, =n\log_{a}x\,

d) \log_{a}a\, =1\,

e) \log_{a}1\, =0\,

Note : All the laws apply for \ln\,/\lg\,

Common Mistakes

Exercise 2

Complete the following table

1 \lg x^{2}y\, 2\lg x + \lg y\,
2 \lg \frac{1}{\left(xy\right)^{2}}\, -2\lg x -2 \lg y\,
3 \lg \sqrt{\frac{xy^{3}}{z^{2}}} \frac{1}{2}\lg x+\frac{3}{2}\lg y-\lg z
4 \lg \left(\frac{x^{2}}{y^{3}}\right) 2\lg x-3\lg y\,
5 \lg \sqrt{xy^{3}} \frac{1}{2}\lg x+\frac{3}{2}\lg y\,
6 \lg \frac{1}{xy^{2}} -\lg x-2\lg y\,

Change of base

f) \log_{a}b\, =\frac{\log_{c}b}{\log_{c}a}(change of base to c\,)

g) \log_{a}b\, =\frac{1}{\log_{b}a}(change of base to b\,)

Examples

Algebra (Revision)

Just something I find some students can get confused/careless at times








Exercise 3

Express the following in terms of u=\log_{2}x\,

1) \log_{2}x^{3}\, =3u\,

2) \log_{2}\frac{1}{\sqrt[3]{x^{2}}}\, =-\frac{2}{3}u\,

3) \log_{2}2x^{3}\, =1+3u\,

4) \log_{2}\frac{x}{4}\, =u-2\,

5) \log_{2}\frac{1}{8x}\, =-3-u\,

6) \log_{2}4\sqrt {x}\, =2+\frac{1}{2}u\,

7) \log_{4}x\, =\frac{1}{2}u\,

8) \log_{\frac{1}{2}}x\, =-u\,

9) \log_{\sqrt{8}}x\, =\frac{2}{3}u\,

10) \log_{\frac{1}{\sqrt{8}}}x\, ==-\frac{2}{3}u\,

11) \log_{4}2x\, =\frac{1+u}{2}

12) \log_{8}\frac{4}{\sqrt{x}}\, =\frac{4+u}{6}

13) \log_{4x}2\, =\frac{1}{2+u}

14) \left(\log_{2}x\right)^{2} =u^{2}\,

Value of a^{\log_{a}x}

h) a^{\log_{a}x} =x\,

i) e^{\ln x}\, =x\,

Note : a^{n\log_{a}x} =a^{\log_{a}x^{n}} =x^{n}\,

Exercise 4

1) 3^{\log_{3}4} =4\,

2) 2^{-\log_{2}3} =\frac{1}{3}

3) 10^{\lg 4}\, =4\,

4) 10^{-2\lg x}\, =\frac{1}{x^{2}}

5) e^{-3\ln 2}\, =\frac{1}{8}

6) e^{-\frac{1}{2}\ln x}\, =\frac{1}{\sqrt{x}}

Same Base

Thus, to solve logarithm equations with same/similar base, we rearrange the equation into

Note : Check that

Examples

Exercise 5

Solve the following equations

1) \log_{2}x=1-\log_{2}8x\, x=\frac{1}{2}\,

2) \lg\left(x-4\right)+\lg\left(x-5\right)=\lg 6\, x=7\,

3) \ln\left(x+2\right)=2+\ln2 x=2e^{2}-2\,

4) \log_{x}16-4\log_{x}3=4\, x=\frac{2}{3}\,

5) \log_{\sqrt{8}}\left(x-5\right)=\frac{2}{3}+\log_{8}\left(x-6\right) x=7\,

6) \frac{1}{2}\log_{3}\frac{x+3}{x-2}=\log_{9}{\left(x-5\right)} x=7\,

7) \log_{6}\frac{x^{2}-5x}{x-9}=0 x=3\,

Using Substitution

When necessary, we use substitution. For example, when we have both \log_{x}3\, & \log_{3}x\,, we know we can change one to the other easily.

Examples

Exercise 6

Solve the following equations

1) \log_{2}x=\log_{x}16\, x=4 \mbox{ or }  x=\frac{1}{4}

2) \log_{x}5+1=6\log_{5}x\, x=\frac{1}{\sqrt[3]{5}} \mbox{ or }  x=\sqrt{5}

3) 2\left(3\log_{x}4-\log_{4}x\right)=1\, x=8 \mbox{ or } x=\frac{1}{16}

4) \log_{x}3+\log_{9}x=\frac{3}{2}\, x=9 \mbox{ or } x= 3\,

5) \log_{2}4x=\log_{x}8\, x=\frac{1}{8} \mbox{ or } x=2 \,

Using Logarithm to Solve Exponent Equations

Writing the final answer

How we write the final answer depends on the question. A careless mistake can result in loss of marks.

Note that we do need to round off

Example

Solve for x\, (3sf)

More Rearranging

Example

Identifying Type of Exponent Equations

As always, the difficulty isn't with the solving, it is deciding which method to use for a particular question.

Exercise 7

Solve the following equations, leaving your answers correct to 3 s.f.

1) 4^{3x-2}=15\, x=1.32\,

2) 3^{x}\cdot2^{x+1}=5\, x=0.511\,

3) 2^{x}\cdot5^{x}=3^{x+2}\, x=1.82\,

4) 3^{x}\cdot4^{1-x}=5^{2x-1}\, x=0.854\,

5) 2^{x+1}\cdot3^{x-2}=4^{x-2}\, x=-3.13\,

6) 4^{3x}\cdot5^{2x-1}=6^{x+2}\, x=0.930\,

7) 2\left(9^{x}\right)-7\left(3^{x}\right)+3=0 x= -0.631 \mbox{ or } x=1\,

8) 2^{x}+1=12\left(2^{-x}\right) x=1.58\,

Solving Simultaneous Equations

In simultaneous equation, the main step is always getting rid of one of the variable. We can do it in a few ways, depending of the situation.

Examples

Exercise 8

1) 3^{x}=9\cdot3^{3-y} \mbox{ and }\log_{4}x+\log_{4}y=1 y=4, x=1 \mbox{ or } y=1, x=4 \,

2) xy=4 \mbox{ and } \log_{2}y-4\log_{2}x=2\, x=1, y=4\,

3) \log_{2}x+\log_{2}y=8 \mbox{ and } \log_{2}x=3\log_{2}y\, x=64 , y=4\,

4) \log_{2}x+\log_{2}y=3 \mbox{ and } \log_{2}\left(\frac{x}{y^{2}}\right)=9\, x=32 , y=\frac{1}{4}

5) \log_{4}xy=1 \mbox{ and } \left(\log_{2}x\right)\left(\log_{2}y\right)=-3\, x=8, y=\frac{1}{2} \mbox { or } x=\frac{1}{2}, y=8

6) \log_{3}xy^{2}=3 \mbox{ and } \left(\log_{3}x\right)\left(\log_{3}y\right)=1\, x=9, y=\sqrt{3} \mbox { or } x=3, y=3

7) \log_{2}x+\log_{2}y=5 \mbox{ and } \log_{y}{x}=\frac{3}{2}\, x=4 , y=8\,

8) 2^{x+2}+2^{y+1}=17 \mbox{ and } 5^{x-1}-5^{y+2}=0\, x=2 , y=-1\,

Simple Inequalities

If a>b\,

IMPORTANT

If a>0\,

Exercise 9

Complete the following table

1 -2x>2\, x <-1\,
2 2x>-2\, x>-1\,
3 -2x>-2\, x<1\,
4 x-2>-2\, x>0\,
5 x\lg 2>\lg 3\, x>\frac{\lg 3}{\lg 2}
6 x\ln \frac{1}{2}>\ln 3\, x<\frac{\ln 3}{\ln \frac{1}{2}}
7 x\ln 0.3< \ln 0.1\, x>\frac{\ln 0.1}{\ln 0.3}
8 x\lg \frac{5}{4}> -\lg 3\, x>-\frac{\lg 3}{\lg \frac{5}{4}}

Solving Inequalities

n integer

If we are asked

For example

Examples

Exercise 10

1) Find the smallest integer n\, such that 3^{n}>25600\, n=10\,

2) Find the smallest integral value of n\, such that \left(0.85\right)^{n}<0.0001\, n=57\,

3) Find the largest integral value of n\, such that 7\left(4^{n}\right)<5000 n=4\,

4) Find the largest integer n\, such that 4\left(\frac{3}{5}\right)^{n+1}>\frac{1}{100} n=10\,

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