Last modified on March 22nd, 2024

chapter outline

 

Logarithmic to Exponential Form

Logarithm is the exact opposite of exponentiation. A number written in the exponent form is much easier to interpret than when it is expressed in the logarithmic form. 

In mathematical calculations, we frequently convert a given logarithmic equation to its corresponding exponential form.

For example, 33 = 27 is much easier to understand compared to the form log3(27) = 3

Formula

The relationship between logarithmic and exponential forms can be represented by the formula:

If logbx = a, then ba = x, for all x > 0 and 0 < b ≠ 1

Here, b = base, a = exponent, and x is the result.

Steps

Let us convert the logarithmic representation of log8(512) = 3 in its exponential form.

Step 1: Identifying the Base, Exponent, and Argument

Here, 

Base = 8

Exponent = 3

Argument = 12 

Step 2: Using the Formula 

83 = 512

Thus, log8(512) = 3 in its exponential form is 83 = 512

Converting Natural Logarithms

The natural logarithm y = ln(x) or y = loge(x) can also be converted to its exponential form as:

If y = ln(x) or y = loge(x), then ey = x for all x > 0.

For example, the exponential form of the natural logarithm y = ln(4.5) ≈ 1.504 is e1.504 ≈ 4.5

Solved Examples

Find the value of log225 and convert it into its exponential form (Given: log5 = 0.699 and log3 = 0.477)

Solution:

As we know, log5 = 0.699 and log3 = 0.477
Here, log225 
= log(52 ⋅ 32
= log(52) + log(32) (by the product rule of logarithm)
= 2 log5 + 2 log 3 (by the power rule of the logarithm)
= 2 × 0.699 + 2 × 0.477
= 1.398 + 0.954
= 2.352
Thus, log225 = 2.352
⇒ 102.352 = 224.905, the required exponential form.

Convert log7(2401) = 4 into its exponential form.

Solution:

As we know, If logbx = a, then ba = x
Here, log7(2401) = 4
⇒ 74 = 2401Thus, the exponential form is 74 = 2401