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Maths grade 11/12, book 1 chapter 14 - Exponentials and logs, natural logs

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Study note

Ahmed Nour ✓™

United States of America

DEFAULT (US)

Math (ACT®)

Exponentials and logarithms

Preparation and review

Exponents and logarithms

exponential functions
f(x) = ax
always go through I
on x-axis
tends towards
0 as
x decreases
graphs of their gradient functions are similar to shape of the actuall function
f(x)
f'(x)
between a= 2 and a=3 → graphs are the SAME
Lo number given as e
f(x)= ex
then f'(x) = ex
f(x) = ekx, then f'(x) = kekx
"
logarithms
logan = x
E
means
= x
So f'(x) = Kf(x)
logax + logay = loga xy
loga x - logay = loga (1)
loga (x²)
клодах
ax=n
natural logarithms
inex
You can take logs of both sides
f(x) = g(x
logf(x) = logg(x)
constant
you can turn equations into linear data
y = axh
logy = logax"
logy = loga + logxn
logy = loga + nlogx → plot logy against logx
y=/ex
-multiplication law.
division law.
power law
eg 3²=9 so log 39 = 2
Ly=x
y=lnx
e²x+3 =
= 7
also
Togaa = 1
loga=0
2x = ln 7-3
x =
logal) = logalx")
= -logax
2x+3
une ²x³ = ln 7°
2x+3 = Un7
2/17 - 12/22
Loas of
both sides  MIXED EXERCISES
1) a) y = 2*
y= 2-x
= (2-¹)² = ²
2) a) loga (²a)
=
Loga p² + loga q
2 loga P + loga q
3) a) p = loyalb
:
몹
=
410992
Logą2
4) a) 4x = 23
x Log4 = log23
x= 10923
1094
5) a) u = 2x
1:4² = 4*
b) y=5e²-1
y = -1
goes through 5
2x+1=2 x 2¹ = 2u
- 4* - 2x+1 -15=0
= U² - 2U-15=0
b) P=logg16
b) loga (Pq) = 5 loga (P²q₂) = 9
loga P + logaq = 5
2 logap + 2loga 9 = 10
2 loga p + loga 9 = 9
logaq =
=1
b) 72x+1=1000
log72x+1
Log 1000
2x+1 log] = log1000
lowers
by one
loga (89) = loga 8 + logq9
3 loga 2 t
4
x = -1/2 x Log 1000 - 12/2
Log7
2x = log 1000 -1
log 7
b) u²-24-15 = 0
u= 5 or -3
= 1.27
- 2* = 5 or -3
log2x = logs
x = logs = 2.32
Tog2
= 2loga P + logaq = 9
c) y =lnx
..
содар=4
r
→x=0
3x² -1 → 38-1
·reflection
of
_y=
ex
can't
have logs of
a negative number
C) 10x = 6*+2
x log10 = (x+2)log b
xlog10x10g6 = 210g6
•xlog6 +21096
x (10g 10-log6) = 210gb
x = 21096 = 7.015
log10-log6

Exponentials and logarithms

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Exponentials and logarithms

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

13 M

#1

950 K+

Still not sure? Look at what your fellow peers are saying...

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

The application is very simple and well designed. So far I have found what I was looking for :D

Love this App ❤️, I use it basically all the time whenever I'm studying