Mathematics I — Week 6: Logarithmic Functions, Properties & Equations

TL;DR

“What exponent gives this number?”

$$\boxed{ a^x=y \iff \log_a y=x }$$ $$a>0,\qquad a\neq1,\qquad y>0.$$

• logs are inverses of exponentials
• logarithm domain is positive reals
• product / quotient / power laws
• change of base
• solving exponential equations
• solving logarithmic equations
• checking domain restrictions
• graph relationship between $a^x$ and $\log_a x$

---

1. What is a logarithm?

$$2^3=8.$$ $$\log_2 8=3.$$

logarithm base $2$ of $8$ equals $3$

$$2^3=8.$$

To what power must the base be raised to obtain the given number?

---

2. Fundamental definition

$$a>0,\qquad a\neq1,$$ $$\boxed{ \log_a x=y \iff a^y=x }$$

---

3. Examples

$$\log_2 8=3$$ $$2^3=8.$$ $$\log_{10}100=2$$ $$10^2=100.$$ $$\log_3\frac19=-2$$ $$3^{-2}=\frac19.$$

---

4. Why logarithms exist

$$x\mapsto a^x.$$ $$a^x\mapsto x.$$ $$\boxed{ \log_a x = (a^x)^{-1} }$$

---

5. Basic restrictions

$$\boxed{ a>0,\quad a\neq1,\quad x>0 }$$ $$\log_a 0$$ $$\log_a(-5)$$

---

6. Domain and range

$$f(x)=\log_a x,$$ $$\boxed{ (0,\infty) }$$ $$\boxed{ \mathbb R }$$ $$a^x: \quad \text{Domain}=\mathbb R, \quad \text{Range}=(0,\infty).$$

---

7. Basic log values

$$a^0=1,$$ $$\boxed{ \log_a1=0 }$$ $$a^1=a,$$ $$\boxed{ \log_a a=1 }$$

---

8. Inverse identities

$$\boxed{ \log_a(a^x)=x }$$ $$\boxed{ a^{\log_a x}=x }$$

---

9. Graph of logarithm

$$a>1,$$ $$y=\log_a x$$ $$(1,0)$$ $$\log_a1=0.$$ $$(a,1).$$

---

10. Exponential/log reflection

$$y=a^x$$ $$y=\log_a x$$ $$\boxed{y=x}.$$ $$(x,y) \leftrightarrow (y,x).$$

---

11. Vertical asymptote

$$y=\log_a x,$$ $$x=0$$ $$x\to0^+,$$ $$\log_a x\to-\infty.$$ $$x=0.$$

---

12. Logarithm growth vs decay

$$a>1,$$ $$\log_a x$$ $$0<a<1,$$ $$\log_a x$$

---

13. Product law

$$\boxed{ \log_a(xy) = \log_a x+\log_a y }$$ $$x=a^m,\qquad y=a^n.$$ $$xy=a^{m+n}.$$ $$\log_a(xy)=m+n.$$ $$m=\log_a x,\qquad n=\log_a y.$$ $$\log_a(xy)=\log_a x+\log_a y.$$

---

14. Quotient law

$$\boxed{ \log_a\left(\frac xy\right) = \log_a x-\log_a y }$$ $$\frac{a^m}{a^n}=a^{m-n}.$$

---

15. Power law

$$\boxed{ \log_a(x^r) = r\log_a x }$$ $$x=a^m$$ $$x^r=a^{mr}.$$

---

16. Important false identities

$$\log(x+y) \neq \log x+\log y$$ $$\log(x-y) \neq \log x-\log y.$$

• multiplication
• division
• powers

---

17. Compressing logarithms

$$\log_a x+\log_a y.$$ $$\boxed{ \log_a(xy) }$$ $$2\log_a x-\log_a y.$$ $$2\log_a x=\log_a(x^2).$$ $$\boxed{ \log_a\left(\frac{x^2}{y}\right) }$$

---

18. Expanding logarithms

$$\log_a\left(\frac{x^3y^2}{z}\right).$$ $$= \log_a(x^3y^2)-\log_a z.$$ $$= \log_a x^3+\log_a y^2-\log_a z.$$ $$\boxed{ 3\log_a x+2\log_a y-\log_a z }$$

---

19. Change-of-base formula

$$\boxed{ \log_a x = \frac{\log_b x}{\log_b a} }$$ $$\ln$$ $$\log_{10}.$$

---

20. Deriving change of base

$$y=\log_a x.$$ $$a^y=x.$$ $$\log_b(a^y)=\log_b x.$$ $$y\log_b a=\log_b x.$$ $$y= \frac{\log_b x}{\log_b a}.$$ $$\boxed{ \log_a x = \frac{\log_b x}{\log_b a} }$$

---

21. Common logarithm

$$10,$$ $$\log x$$ $$\log_{10}x.$$

---

22. Natural logarithm

$$e,$$ $$\boxed{ \ln x }$$ $$\log_e x.$$ $$e\approx2.71828.$$

---

23. Why $e$ is special

$$e^x$$ $$\ln x.$$ $$\ln(e^x)=x$$ $$e^{\ln x}=x.$$

---

24. Solving exponential equations: same base

$$2^{x+1}=16.$$ $$16=2^4.$$ $$2^{x+1}=2^4.$$ $$x+1=4.$$ $$\boxed{x=3}$$

---

25. Another same-base example

$$9^x=27.$$ $$9=3^2$$ $$27=3^3.$$ $$(3^2)^x=3^3.$$ $$3^{2x}=3^3.$$ $$2x=3$$ $$\boxed{x=\frac32}$$

---

26. Exponential equations without same base

$$2^x=7.$$ $$\log(2^x)=\log7.$$ $$x\log2=\log7.$$ $$\boxed{ x= \frac{\log7}{\log2} }$$ $$\log_2 7.$$

---

27. General exponential equation

$$a^x=b,$$ $$\boxed{ x=\log_a b }$$ $$\boxed{ x= \frac{\ln b}{\ln a} }$$

---

28. Exponential equation with coefficient

$$3\cdot2^x=24.$$ $$2^x=8.$$ $$x=3.$$

---

29. Example with exponent expression

$$5^{2x-1}=11.$$ $$\ln(5^{2x-1})=\ln11.$$ $$(2x-1)\ln5=\ln11.$$ $$2x-1=\frac{\ln11}{\ln5}.$$ $$\boxed{ x= \frac12 \left( 1+\frac{\ln11}{\ln5} \right) }$$

---

30. Solving logarithmic equations

$$\log_2 x=5.$$ $$x=2^5.$$ $$\boxed{x=32}$$

---

31. Example

$$\log_3(x-1)=2.$$ $$x-1=3^2.$$ $$x-1=9$$ $$\boxed{x=10}$$ $$x-1>0.$$

---

32. Log equation using product law

$$\log_2 x+\log_2(x-2)=3.$$ $$x>0$$ $$x-2>0.$$ $$x>2.$$ $$\log_2[x(x-2)]=3.$$ $$x(x-2)=2^3=8.$$ $$x^2-2x-8=0.$$ $$(x-4)(x+2)=0.$$ $$x=4,\quad -2.$$ $$x>2.$$ $$\boxed{x=4}$$

---

33. Extraneous solutions matter

$$\boxed{ \text{every logarithm argument must be }>0 }$$

---

34. Example with rejected root

$$\log(x-1)+\log(x-3)=\log3.$$ $$x>3.$$ $$\log[(x-1)(x-3)]=\log3.$$ $$(x-1)(x-3)=3.$$ $$x^2-4x+3=3.$$ $$x^2-4x=0.$$ $$x(x-4)=0.$$ $$x=0,\quad4.$$ $$x>3.$$ $$\boxed{x=4}$$

---

35. Equality of logarithms

$$\log_a u=\log_a v$$ $$\boxed{u=v}$$ $$a^u=a^v\Rightarrow u=v.$$

---

36. Example

$$\log_5(x+2)=\log_5(3x-4).$$ $$x+2=3x-4.$$ $$6=2x$$ $$x=3.$$ $$x+2=5>0$$ $$3x-4=5>0.$$

---

37. Graph transformations

$$y=\log_a x.$$ $$y=\log_a(x-h)+k$$

• right by $h$
• up by $k$

$$x=0$$ $$\boxed{x=h}.$$

---

38. Example

$$f(x)=\log_2(x-3)+5.$$ $$x-3>0.$$ $$\boxed{x>3}$$ $$\boxed{x=3}$$ $$\mathbb R.$$

---

39. Finding domain of logarithmic expressions

$$f(x)=\log(x+4).$$ $$x+4>0.$$ $$\boxed{x>-4}$$

---

40. Multiple factors inside log domain

$$f(x)=\log[(x-1)(x+3)].$$ $$(x-1)(x+3)>0.$$ $$x=-3,\quad1.$$ $$\boxed{ x<-3 \quad\text{or}\quad x>1 }$$

Each factor does not individually need to be positive.

---

41. Logarithm of reciprocal

$$\log_a\left(\frac1x\right) = \log_a1-\log_a x.$$ $$\log_a1=0,$$ $$\boxed{ \log_a\left(\frac1x\right) = -\log_a x }$$

---

42. Negative exponents and logs

$$a^{-x}=\frac1{a^x},$$ $$\log_a(a^{-x})=-x.$$

---

43. Log of roots

$$\sqrt[n]{x}=x^{1/n},$$ $$\boxed{ \log_a\sqrt[n]{x} = \frac1n\log_a x }$$ $$\log_2\sqrt8 = \frac12\log_2 8 = \frac32.$$

---

44. Exponential growth model

$$P(t)=P_0a^t$$ $$P(t)=P_0e^{kt}.$$

• $P_0$ = initial amount
• $t$ = time
• $k$ = growth/decay parameter

---

45. Example: solving for time

$$P(t)=100(2)^t.$$ $$P(t)=800?$$ $$100(2)^t=800.$$ $$2^t=8.$$ $$\boxed{t=3}$$

---

46. General growth-time equation

$$P(t)=P_0e^{kt}.$$ $$\frac{P(t)}{P_0}=e^{kt}.$$ $$\ln\left(\frac{P(t)}{P_0}\right)=kt.$$ $$\boxed{ t= \frac1k \ln\left(\frac{P(t)}{P_0}\right) }$$

---

47. Why logs are useful conceptually

multiplication → addition

$$\log(xy)=\log x+\log y$$

division → subtraction

$$\log(x/y)=\log x-\log y$$

exponentiation → multiplication

$$\log(x^r)=r\log x$$

---

48. Orders of magnitude

$$10^3=1000$$ $$\log_{10}(1000)=3.$$ $$10^6=1,000,000$$ $$\log_{10}(1,000,000)=6.$$ $$10^3=1000.$$

---

$$\log_a0$$ $$0.$$ $$\log_a1=0.$$

---

$$\log_a1=0$$ $$\log_1a$$ $$1$$

---

$$\log(xy)=\log x+\log y.$$ $$\log(x+y)=\log x+\log y.$$ $$\log(x+y).$$

---

$$\log(x^3)=3\log x.$$ $$\log(x^2)$$ $$\log x$$

---

$$\boxed{ a>0,\qquad a\neq1 }$$

---

$$\log(x-5)=\log(2-x),$$ $$x-5>0 \Rightarrow x>5$$ $$2-x>0 \Rightarrow x<2.$$ $$\boxed{\text{no real solution}}$$

---

55. A useful solving workflow

text
1. Write domain restrictions
2. Simplify/merge logarithms
3. Convert log form ↔ exponential form
4. Solve resulting algebra
5. Check every candidate against original domain

text
1. Isolate exponential term
2. Try rewriting both sides with same base
3. If impossible, take logs
4. Use power law
5. Solve algebraically

---

56. Full example: exponential equation

$$4^{x+1}=7.$$ $$\ln(4^{x+1})=\ln7.$$ $$(x+1)\ln4=\ln7.$$ $$x+1=\frac{\ln7}{\ln4}.$$ $$\boxed{ x= \frac{\ln7}{\ln4}-1 }$$

---

57. Full example: logarithmic equation

$$\log_3(x+1)+\log_3(x-1)=2.$$

Domain

$$x+1>0$$ $$x-1>0.$$ $$x>1.$$

Combine

$$\log_3[(x+1)(x-1)]=2.$$

Convert

$$(x+1)(x-1)=3^2.$$ $$x^2-1=9.$$ $$x^2=10.$$ $$x=\pm\sqrt{10}.$$ $$x>1.$$ $$\boxed{x=\sqrt{10}}$$

---

58. Exponential/log symmetry table

---

59. Formula sheet

Definition

$$\boxed{ \log_a x=y \iff a^y=x }$$

Product

$$\boxed{ \log_a(xy)=\log_a x+\log_a y }$$

Quotient

$$\boxed{ \log_a\frac xy = \log_a x-\log_a y }$$

Power

$$\boxed{ \log_a(x^r)=r\log_a x }$$

Change of base

$$\boxed{ \log_a x = \frac{\log_bx}{\log_ba} }$$

Inverse identities

$$\boxed{ \log_a(a^x)=x }$$ $$\boxed{ a^{\log_a x}=x }$$

---

60. Weeks 5–6 as one idea

$$f(x)=a^x$$ $$f^{-1}(x)=\log_a x.$$ $$\boxed{ x \overset{a^x}{\longrightarrow} a^x \overset{\log_a}{\longrightarrow} x }$$

---

61. What you should actually master

1. Translate between: $$a^y=x$$ and: $$\log_a x=y.$$
2. Know log domain/base restrictions.
3. Use:
   • product law
   • quotient law
   • power law.
4. Expand and condense logarithmic expressions.
5. Apply change of base.
6. Recognize exponential/log graphs as inverses.
7. Identify domain, range and asymptotes.
8. Solve same-base exponential equations.
9. Solve arbitrary-base exponential equations using logs.
10. Solve basic logarithmic equations.
11. Combine multiple logarithms before solving.
12. Reject extraneous/domain-invalid answers.
13. Work with $\ln$ and $e$ at a basic level.

Next

$$\boxed{ \text{finite algebra} \rightarrow \text{behavior as something approaches a value} }$$

---

You

---

ChatGPT

---

Quick Recall & Flashcards

• Definition of $\log_b(x) = y$ :: $b^y = x$ ($x > 0, b > 0, b \neq 1$)
• Product rule for logarithms: $\log_b(xy)$ :: $\log_b(x) + \log_b(y)$
• Quotient rule for logarithms: $\log_b(x/y)$ :: $\log_b(x) - \log_b(y)$
• Power rule for logarithms: $\log_b(x^k)$ :: $k \log_b(x)$
• Change of base formula for $\log_b(a)$ :: $\frac{\ln(a)}{\ln(b)} = \frac{\log_{10}(a)}{\log_{10}(b)}$
• Natural logarithm $\ln(x)$ base :: Base $e \approx 2.71828$
• Dangerous trap when solving log equations :: Must check for extraneous roots (log argument must be $> 0$)

---

Practice & Self-Check