Ximpledu

Logarithm

See how to solve a logarithm
(expression/equation/inequality/function).
33 examples and their solutions.

Logarithmic Form

Definition

2m = 3
m = log2 3
Logarithm (log) is a way
to write the exponent of a number.
log2 3 is read as
[log base 2 of 3].
Exponent Rules

Example

24 = 16
→ Logarithmic form?
Solution

Example

3-2 = 19
→ Logarithmic form?
Solution

Example

512 = √5
→ Logarithmic form?
Solution

Example

2 = log3 9
→ Exponential form?
Solution

Example

-5 = log2 132
→ Exponential form?
Solution

Example

23 = log7 349
→ Exponential form?
Solution

Logarithm of 1

Formula

loga 1 = 0

Logarithm of Itself

Formula

loga a = 1

loga xm

Formula

loga xm
= m loga x

Example

log2 8
Solution

Example

log3 181
Solution

Example

log3 2 = a
log3 32 = ?
Solution

loga xy

Formula

loga xy
= loga x + loga y

Example

log2 3 = a
log2 24 = ?
Solution

loga xy

Formula

loga xy
= loga x - loga y

Example

log2 328
Solution

Example

log6 9 - log6 15 + log6 10
Solution

Logarithmic Equation

Formula

loga x

x > 0
0 < a < 1, a > 1
x and a should satisfy these conditions.

Example

log3 x = 4
Solution

Example

logx 64 = 3
Solution

Example

log2 (log3 (log5 x)) = 0
Solution

Example

log2 x + log2 (x - 1) = log2 12
Solution

Example

(log5 x)2 - log5 x2 - 3 = 0
Solution

Logarithmic Inequality

Example

log7 (x + 2) ≤ 1
Solution

Example

log0.1 (x - 3) > 2
Solution

Example

2 log3 x ≥ log3 (x + 6) + 1
Solution

Logarithmic Function: Graph

Graph: y = loga x (a > 1)

1. The graph passes (1, 0).
2. The asymptote of the graph is the y-axis.
(= The graph follows the y-axis.)
So the domain x is always (+): x > 0.

Graph: y = loga x (0 < a < 1)

1. The graph passes (1, 0).
2. The asymptote of the graph is the y-axis.
(= The graph follows the y-axis.)
So the domain x is always (+): x > 0.

Graph: y = ax and y = loga x

y = ax and y = loga x
are symmetric about y = x.

So these two function are inverse functions.
Exponential Function: Graph

Example

Graph y = log3 x.
Solution

Example

Graph y = log2 (x - 3).
Solution

Common Logarithm

Definition

log x = log10 x
A common logarithm (common log)
is a logarithm
whose base is 10.
In high school math,
log10 = log.

Example

log 5 = ?
(Assume log 2 = 0.301.)
Solution

Example

log 120 = ?
(Assume log 2 = 0.301, log 3 = 0.477.)
Solution

Change of Base Formula

Formula

loga x = logb xlogb a

Example

log2 70 = ?
(Assume log 2 = 0.301, log 7 = 0.845.)
Solution

Example

log2 3 = a
log12 18 = ?
Solution

Example

(log2 27)(log9 16)
Solution

Natural Logarithm

Definition

ln x = loge x
A natural logarithm (natural log)
is a logarithm
whose base is e.
(e is a constant number.
e = 2.71828...)
In high school math,
loge = ln.

Example

ln 2ex + 1 = ?
(Assume ln 2 = 0.69.)
Solution

Exponential Growth/Decay: Time

Formula

A = A0(1 + r)t
A: final value
A0: initial value
r: rate of change
t: time
By using logarithm,
you can find the time t.
Exponential Growth/Decay: Final Value
Compound Interest

Example

The population of a town is 10,000. If it increases at a rate of 8% per year, after how many years will the population be more than 24,000?
(Assume log 2.4 = 0.380, log 1.08 = 0.033.)
Solution

Example

A radioactive substance weighs 100g. If it decreases at a rate of 14% per week, after how many weeks will the weight be less than 10g?
(Assume log 0.86 = -0.066.)
Solution

Continuous Exponential Growth/Decay: Time

Formula

A = A0ert
A: final value
A0: initial value
e: constant number (= 2.71828...)
r: rate of change
t: time
Continuous Exponential Growth/Decay: Final Value
Compound Interest

Example

A substance weighs 10g. If it continuously increases at a rate of 5% per minute, after how many minutes will the weight be more than 30g?
(Assume ln 3 = 1.099.)
Solution

Example

A radiocative substance weighs 50g. If it continuously decreases at a rate of 3% per second, after how many seconds will the weight be less than 20g?
(Assume ln 0.4 = -0.916.)
Solution

Half-Life

Formula

-rt = ln 2

r: rate of change
t: half-life
The half time is the amount of time
when a value
continuously & exponentially decreases
to one half.
A0 → A0/2
The formula came from
A0ert = A = A0/2.
Continuous Exponential Growth/Decay: Time

Example

A weight of a radioactive substance is continuously decreasing at a rate of 4% per second. Find the half-life of the substance.
(Assume ln 2 = 0.69.)
Solution