Convert Exponentials and Logarithms



In convert Exponentials and Logarithms we will mainly discuss how to change the logarithm expression to Exponential expression and conversely from Exponential expression to logarithm expression.

To discus about convert Exponentials and Logarithms we need to first recall about logarithm and exponents.

The logarithm of any number to a given base is the index of the power to which the base must be raised in order to equal the given number. Thus, if aˣ = N, x is called the logarithm of N to the base a.

For example:

1. Since 3⁴ = 81, the logarithm of 81 to base 3 is 4.

2. Since 10¹ = 10, 10² = 100, 10³ = 1000, ………….

The natural number 1, 2, 3, …… are respectively the logarithms of 10, 100, 1000, …… to base 10.

The logarithm of N to base a is usually written as log₀ N, so that the same meaning is expressed by the two equations

ax = N; x = loga N



Examples on convert Exponentials and Logarithms

1. Convert the following exponential form to logarithmic form:

(i) 104 = 10000

Solution:

104 = 10000

⇒ log10 10000 = 4


(ii) 3-5 = x

Solution:

3-5 = x

⇒ log3 x = -5


(iii) (0.3)3 = 0.027

Solution:

(0.3)3 = 0.027

⇒ log0.3 0.027 = 3



2. Convert the following logarithmic form to exponential form:

(i) log3 81 = 4

Solution:

log3 81 = 4

⇒ 34 = 81, which is the required exponential form.


(ii) log8 32 = 5/3

Solution:

log8 32 = 5/3

⇒ 85/3 = 32


(iii) log10 0.1 = -1

Solution:

log10 0.1 = -1

⇒ 10-1 = 0.1.


3. By converting to exponential form, find the values of following:

(i) log2 16

Solution:

Let log2 16 = x

⇒ 2x = 16

⇒ 2x = 24

⇒ x = 4,

Therefore, log2 16 = 4.


(ii) log3 (1/3)

Solution:

Let log3 (1/3) = x

⇒ 3x = 1/3

⇒ 3x = 3-1

⇒ x = -1,

Therefore, log3(1/3) = -1.


(iii) log5 0.008

Solution:

Let log5 0.008 = x

⇒ 5x = 0.008

⇒ 5x = 1/125

⇒ 5x = 5-3

⇒ x = -3,

Therefore, log5 0.008 = -3.


4. Solve the following for x:

(i) logx 243 = -5

Solution:

logx 243 = -5

⇒ x-5 = 243

⇒ x-5 = 35

⇒ x-5 = (1/3)-5

⇒ x = 1/3.


(ii) log√5 x = 4

Solution:

log√5 x = 4

⇒ x = (√5)4

⇒ x = (51/2)4

⇒ x = 52

⇒ x = 25.


(iii) log√x 8 = 6

Solution:

log√x 8 = 6

⇒ (√x)6 = 8

⇒ (x1/2)6 = 23

⇒ x3 = 23

⇒ x = 2.


Logarithmic Form Vs. Exponential Form

The logarithm function with base a has domain all positive real numbers and is defined by

loga M = x           ⇔           M = ax

                                                                 where M > 0, a > 0, a ≠ 1


Logarithmic Form                          Exponential Form

loga M = x           ⇔           M = ax


Log7 49 = 2          ⇔           72 = 49


Write the exponential equation in logarithmic form.


Exponential Form                          Logarithmic Form


M = ax          ⇔           loga M = x

24 = 16          ⇔           log2 16 = 4

10-2 = 0.01          ⇔           log10 0.01 = -2

81/3 = 2          ⇔           log8 2 = 1/3

6-1 = 1/6          ⇔           log6 1/6 = -1


Write the logarithmic equation in exponential form.


Logarithmic Form                          Exponential Form


loga M = x           ⇔           M = ax


log2 64 = 6          ⇔           26 = 64


log4 32 = 5/2          ⇔          45/2= 32


log1/82 = -1/3          ⇔           (1/8)-1/3 = 2


log3 81 = x          ⇔           3x = 81


log5 x = -2          ⇔           5-2 = x


log x = 3          ⇔          103 = x




Solve for x:


1. log5 x = 2

x = 52

= 25


2. log81 x = ½

x = 811/2

⇒ x= (92)1/2

⇒ x = 9


3. log9 x = -1/2

x = 9-1/2

⇒ x = (32)-1/2

⇒ x = 3-1

⇒ x= 1/3


4. log7 x = 0

x= 70

⇒ x = 1


Solve for n:


1. log3 27 = n

3n = 27

⇒ 3n = 33

⇒ n = 3


2. log10 10,000 = n

10n = 10,000

⇒ 10n = 104

⇒ n = 4


3. log49 1/7 = n

49n = 1/7



⇒ (72)n = 7-1

⇒ 72n = 7-1

⇒ 2n = -1

⇒ n = -1/2


4. log36 216 = n

36n = 216

⇒ (62)n = 63

⇒ 62n= 63

⇒ 2n = 3

⇒ n = 3/2


Solve for b:



1. logb 27 = 3

b3 = 27

⇒ b3 = 33

⇒ b = 3


2. logb 4 = 1/2

b1/2 = 4

⇒ (b1/2)2 = 42

⇒ b = 16


3. logb 8 = -3

b-3 = 8 ⇒ b-3 = 23

⇒ (b-1)3 = 23

⇒b-1 = 2

⇒ 1/b = 2

⇒ b = ½


4. logb 49 = 2

b2 = 49

⇒ b2 = 72

⇒ b = 7

If f(x) = log3 x, find f(1).

Solution:


f(1) = log3 1 = 0 (since logarithm of 1 to any finite non-zero base is zero.)

Therefore f(1) = 0

A number that is domain of the function y = log10 x is

(a) 1

(b) 0

(c) ½

(d) =10

Answer: (b)


The graph of y = log4 x lines entirely in quadrants

(a) I and II

(b) II and III

(c) I and III

(d) I and IV


At what point does the graph of y = log5 x intersect the x-axis?

(a) (1, 0)

(b) (0, 1)

(c) (5, 0)

(d) There is no point of intersection.

Answer: (a)

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