Logarithm of an exponential term

Logarithm of an exponential term The system of logarithm is introduced in mathematics based on principle of the exponentiation. Exponential form terms are appeared in logarithmic system. So, a logarithmic identity is essential to simply such expressions. Actually, logarithm of an exponential term is equal to product of exponent and logarithmic of the base and it is proved here in algebraic form.

Proof

Assume, b, x and m are three algebraic factors. Assume, the value of b raised to the power x is equal to m and it is expressed in algebraic form. According to logarithmic system, the relation between an exponential term bx and m can also be written in logarithmic form.

bx = m x = logbm

Assume, n is another algebraic factor and it has a relation with algebraic factor m in exponential form. In other words, mn.

mn

Actually, the value of m is equal to bx. So, replace m by its original value.

mn = (bx)n

According to power rule of exponentiation, exponent of an exponential term is equal to product of exponents of same base.

mn = bx×n

Assume, x×n = y and mn = z

mn = bx×n

z = by

Apply principle of logarithm to this exponential term.

by = z y = logbz

Now, replace the algebraic factors y and z by their actual values.

y = logbz

x×n = logbmn

Now, replace the algebraic factor x by its actual value.

(logbm)×n = logbmn

It can be written as follows.

logbmn = n × logbm

It is proved that logarithm of an exponential term is equal to product of exponent and logarithm of same base.

Example

34

It is an exponential form term. Logarithm of exponential term can be found by using the logarithmic system.

loge34 = loge 81

loge34 = 4.394449155

Now, calculate the 4 times of the logarithm of 3.

4 × loge 3 = 4 × 1.098612289

4 × loge 3 = 4.394449155

Therefore,

loge34 = 4 × loge 3 = 4.394449155

The value of logarithm of an exponential term is equal to product of the exponent and logarithm of the same base.

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