Chapter 6: Logarithms

Question 1

Use logarithms as a way to solve exponential equations
(when they cannot be equated to the same base)

Answer 1

Logarithms: inverse of the exponential function (base 10 or base a)
- Base: base of exponent
- Logarithm: power that base must be raised to to obtain that number
Eg. log1000=3, as 10³=1000
Eg. logₐb=x, aˣ=b

Therefore:
log10ˣ=x
x=10ˡᵒᵍˣ

Question 2

State the 5 laws of logarithms

Answer 2

1) logₐm + logₐn = logₐ(mn)
2) logₐm-logₐn = logₐ(m/n)
3) logₐ(bᵐ) = m logₐb
4) logₐ1 = 0
5) logₐa=1

Eg. log₂8 + log₂16
=log₂(8x16)
= log₂(128)
= log₂(2⁷)
= 7log₂2
=7

OR
=log₂(2³) + log₂(2⁴)
= 3log₂2+4log₂2
= 7

Question 3

Define natural logarithms and state the laws of natural logarithms

Answer 3

Natural logarithm (ln): when the base of log is e
(logₑx=lnx)

Laws of ln:
1) ln(m) + ln(n) = ln (mn)
2) ln(m) - ln(n) = ln (m/n)
3) ln(mⁿ)=nln(m)
4) lne=1
4) ln1=0

Question 4

Solve logarithmic equations

Eg. log₂m = 3log₂a - 2

Answer 4

Eg.
log₂m = 3log₂a - 2
log₂m = log₂a³ - log₂2²
log₂m = log₂(a³/4)
m = a³/4

Question 5

State and apply the change of base rule

Answer 5

logₐb = log꜀b/log꜀a

Eg. log₂9 = log9/log2

Question 6

Explain the relationship between exponential (aˣ) and logarithmic functions (logₐx)

Answer 6

Logarithmic functions are the inverse of exponential functions:
- Reflected at y=x
- Domain of logₐx is the range of aˣ
(x>0 –> y>0)
- Range of logₐx is the domain of aˣ
(y∈R –> x∈R)

• Asymptote of logₐx is x=0
  (Asymptote of aˣ is y=0)
• x-intercept is 1
  (when logₐx = 0, x=1)