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 log_bx = a, then b^a = 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 = log_e(x) can also be converted to its exponential form as:

If y = ln(x) or y = log_e(x), then e^y = x for all x > 0.

For example, the exponential form of the natural logarithm y = ln(4.5) ≈ 1.504 is e^1.504 ≈ 4.5

Solved Examples