Math Doubts

Power Rule of Logarithm of an Exponential term to another term

Formula

$\Large \log_{b^y} m^x = \Big(\dfrac{x}{y}\Big) \log_{b} m$

Proof

$m$ and $b$ are two literal numbers and logarithm of $m$ to $b$ is written as $\log_{b} m$ in mathematics.

$x$ and $y$ are also two literals and they are exponents of literals $m$ and $b$ respectively. Then the logarithm of $m$ raised to the power of $x$ to $n$ raised to the power of $y$ is expressed in mathematics as follows.

$\large \log_{b^y} m^x$

It can be transformed as $\log_{b} m$.

Step: 1

Take $t = b^y$ and the logarithmic function can be written as follows.

$\implies \large \log_{b^y} m^x = \log_{t} m^x$

According to the Power Rule of logarithm of an exponential term to a number is equal to the product of exponent of the number and logarithm of the number to the base.

$\implies \large \log_{b^y} m^x = x \log_{t} m$

Replace the actual value of the base $t$.

$\implies \large \log_{b^y} m^x = x \log_{b^y} m$

Step: 2

As per the Power Rule of logarithm of a number to an exponential term is equal to the product of the reciprocal of the exponent and the logarithm of the number to the base of the exponential term.

$\implies \large \log_{b^y} m = \dfrac{1}{y} \log_{b} m$

Step: 3

Consider the equation of the first step and replace $\large \log_{b^y} m$ by its value obtained in the second step.

$\implies \large \log_{b^y} m^x = x \log_{b^y} m$

$\implies \large \log_{b^y} m^x = x \times \dfrac{1}{y} \log_{b} m$

$\therefore \,\,\,\,\,\, \large \log_{b^y} m^x = \dfrac{x}{y} \times \log_{b} m$

It is proved that the logarithm of an exponential term to another exponential term is equal to the product of the quotient of the exponent of the number by the exponent of the base and logarithm of the base of the number to base of the exponential term.

Verification

$\large 2^5$ and $\large 3^5$ are two exponential terms. The logarithm of $\large 2^5$ to base $\large 3^5$ is an example to verify this power rule mathematically.

Step: 1

Calculate the value of logarithm of $\large 2^5$ to $\large 3^4$.

$\large \log_{3^4} 2^5$ $=$ $\large \log_{81} 32 = 0.78866$

Step: 2

Now, calculate the right hand side of the power rule.

$\large \dfrac{5}{4} \log_{3} 2 =$ $\large 1.25 \times 0.63093 = 0.78866$

Step: 3

Compare both values of the power law and you obverse that they values of both sides are equal.

$\large \log_{3^4} 2^5$ $=$ $\large \dfrac{5}{4} \log_{3} 2 = 0.78866$

The example has successfully proved the power rule of logarithm of an exponential term to another exponential function.



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