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 a^x = 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 log_a N, so that the same meaning is expressed by the two equations

a^x = N; x = log_a 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

⇒ log_0.3 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

⇒ 8^5/3 = 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

⇒ 2^x = 16

⇒ 2^x = 2⁴

⇒ x = 4,

Therefore, log₂ 16 = 4.

(ii) log₃ (1/3)

Solution:

Let log₃ (1/3) = x

⇒ 3^x = 1/3

⇒ 3^x = 3^-1

⇒ x = -1,

Therefore, log₃(1/3) = -1.

(iii) log₅ 0.008

Solution:

Let log₅ 0.008 = x

⇒ 5^x = 0.008

⇒ 5^x = 1/125

⇒ 5^x = 5^-3

⇒ x = -3,

Therefore, log₅ 0.008 = -3.

4. Solve the following for x:

(i) log_x 243 = -5

Solution:

log_x 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 = (5^1/2)⁴

⇒ x = 5²

⇒ x = 25.


(iii) log_√x 8 = 6

Solution:

log_√x 8 = 6

⇒ (√x)⁶ = 8

⇒ (x^1/2)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

log_a M = x           ⇔           M = a^x

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


Logarithmic Form                          Exponential Form
log_a M = x           ⇔           M = a^x

Log₇ 49 = 2          ⇔           7² = 49

● Write the exponential equation in logarithmic form.

Exponential Form                          Logarithmic Form

M = a^x          ⇔           log_a M = x

2⁴ = 16          ⇔           log₂ 16 = 4

10^-2 = 0.01          ⇔           log₁₀ 0.01 = -2

8^1/3 = 2          ⇔           log₈ 2 = 1/3

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

● Write the logarithmic equation in exponential form.

Logarithmic Form                          Exponential Form

log_a M = x           ⇔           M = a^x

log₂ 64 = 6          ⇔           2⁶ = 64

log₄ 32 = 5/2          ⇔          4^5/2= 32

log_1/82 = -1/3          ⇔           (1/8)^-1/3 = 2

log₃ 81 = x          ⇔           3^x = 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 = 81^1/2

⇒ 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. log_b 27 = 3

b³ = 27

⇒ b³ = 3³

⇒ b = 3

2. log_b 4 = 1/2

b^1/2 = 4

⇒ (b^1/2)² = 4²

⇒ b = 16

3. log_b 8 = -3

b^-3 = 8 ⇒ b^-3 = 2³

⇒ (b^-1)³ = 2³

⇒b^-1 = 2

⇒ 1/b = 2

⇒ b = ½


4. log_b 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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