Question

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Answer 1

there are two questions

Answer 2

for #57
start by finding \(f(g(x))\)

Answer 3

\(f(x) = \sqrt{x}\)

\(g(x) = 7x+b\)

\(f(g(x)) = ? \)

Answer 4

do u knw how to find it ha ? :)

Answer 5

u mean g(4) u got 28+b ?

Answer 6

nice :)

Answer 7

keep going...

Answer 8

why 4 is there on right side bottom ?

Answer 9

NOOO

Answer 10

ok

Answer 11

g(4) is NOT same as g*4

Answer 12

g(4) means, function g, evaluated at 4

Answer 13

its a notation thing ok

Answer 14

\[6 = \sqrt{28+b}\]

Answer 15

Yes ! u got it
square both sides, and solve b

Answer 16

\[(6)^2 = (\sqrt{28+b})^2\]
\[36 = 28+b\]
\[36-28 = b\]
\[b=8\]

Answer 17

Excellent !

Answer 18

can we do the next one that is of log

Answer 19

for left side, use below log property :-

\(\huge \log_b x - \log_b y = \log_b \frac{x}{y}\)

Answer 20

so i got\[\log_{2}8 \]

Answer 21

given equation :-

\(\large \log_2 24 - \log_2 3 = \log_5 x\)

apply the log property on left side

given equation :-

\(\large \log_2 \frac{24}{3} = \log_5 x\)

\(\large \log_2 8 = \log_5 x\)

\(\large \log_2 2^3 = \log_5 x\)

Answer 22

yes, next use another log property :-

\(\huge \log_b x^n = n\log_b x\)

Answer 23

\[\large \log_2 2^3 = \log_5 x \]

how did we got this i mean\[\log_{2}2^3 \]

Answer 24

given equation :-

\(\large \log_2 24 - \log_2 3 = \log_5 x\)

apply the log property on left side

given equation :-

\(\large \log_2 \frac{24}{3} = \log_5 x\)

\(\large \log_2 8 = \log_5 x\)

\(\large \log_2 2^3 = \log_5 x\)

apply another log property

\(\large 3\log_2 2 = \log_5 x\)

Answer 25

8 = 2*2*2 = 2^3

Answer 26

that i know but what made us take the cube of 2

Answer 27

u wil see it shortly

Answer 28

next we use another log property

\(\huge \log_b b = 1\)

Answer 29

given equation :-

\(\large \log_2 24 - \log_2 3 = \log_5 x\)

apply the log property on left side

given equation :-

\(\large \log_2 \frac{24}{3} = \log_5 x\)

\(\large \log_2 8 = \log_5 x\)

\(\large \log_2 2^3 = \log_5 x\)

apply another log property

\(\large 3\log_2 2 = \log_5 x\)

apply another log property

\(\large 3*1 = \log_5 x\)

\(\large 3 = \log_5 x\)

Answer 30

Fine so far ? :)

Answer 31

ok

Answer 32

next use below to change log to exponent :-

\(\huge a = \log_b x \)

\(\huge \implies b^a = x\)

Answer 33

given equation :-

\(\large \log_2 24 - \log_2 3 = \log_5 x\)

apply the log property on left side

given equation :-

\(\large \log_2 \frac{24}{3} = \log_5 x\)

\(\large \log_2 8 = \log_5 x\)

\(\large \log_2 2^3 = \log_5 x\)

apply another log property

\(\large 3\log_2 2 = \log_5 x\)

apply another log property

\(\large 3*1 = \log_5 x\)

\(\large 3 = \log_5 x\)

\(5^3 = x\)

\(125 = x\)

Answer 34

where would log go

Answer 35

see my reply before that

Answer 36

ok

Answer 37

we can change log to exponent :-
if \(\large a = \log_b x\), then

\(\large b^a = x\)

Answer 38

u need to knw below to be able to solve similar problems involving logs :-

\(\large \color{red}{ \log_b x + \log_b y = \log_b xy }\)

\(\large \color{red}{ \log_b x - \log_b y = \log_b \frac{x}{y} }\)

\(\large \color{red}{ \log_b x^n = n \log_b x }\)

\(\large \color{red}{ \log_b b = 1 }\)

\(\large \color{red}{ a = \log_b x } \text{ means } \color{red}{ b^a = x } \)

Answer 39

the last property is very powerful, as it allows u to jump between exponent/log

Answer 40

ok

Answer 41

in our problem we have used all above properties except the first one.

in most log problems, u will be using all above 5 properties.
so memorize them ok

Answer 42

if for example it would have been \[ \log_{2}4 \]
then \[\log_{2}2^2 \]

Answer 43

simplify further

Answer 44

\(\log_2 4 = \log_2 2^2 = 2 \log_2 2 = 2*1 = 2 \)

Answer 45

ive used the fourth property : \(\log_b b = 1\)

Answer 46

what ?

Answer 47

i mean just suppose if it \[\log_{2}2 \] in the follow in question the instead of 5^3 it would have been 5^2 right

Answer 48

u mean if we have :-

\(\large 3 = \log_2 x\)

then how to change it to exponent ?

Answer 49

are u asking that ?

Answer 50

if so,

\(\large 3 = \log_2 x \)

\(\large 2^3 = x \)

Answer 51

it changes like that

Answer 52

i meant was
\[\log_{2}4 \]
\[\log_{2}2^2 \]
\[2 =\log_{5}x \]
\[5^2 =\log_{x} \]

Answer 53

i dont get how \(\log_5 x\) suddenly at 3rd step :|

but i see u got the concept correctly :)
keep doing log problems, u wil get familiar wid them in no time..... good luck !