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Last modified on March 22nd, 2024

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, 3^{3} = 27 is much easier to understand compared to the form log_{3}(27) = 3

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

If log_{b}x = a, then b^{a} = x, for all x > 0 and 0 < b ≠ 1

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

Let us convert the logarithmic representation of log_{8}(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 **

8^{3} = 512

Thus, log_{8}(512) = 3 in its exponential form is 8^{3} = 512

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

**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(5^{2} ⋅ 3^{2})

= log(5^{2}) + log(3^{2}) (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

⇒ 10^{2.352} = 224.905, the required exponential form.

**Convert log _{7}(2401) = 4 into its exponential form.**

Solution:

As we know, If log_{b}x = a, then b^{a} = x

Here, log_{7}(2401) = 4

⇒ 7^{4} = 2401Thus, the exponential form is 7^{4} = 2401