You have studied the concept of logarithm with basic example to find the log of any quantity on the basis of another quantity. Here is some more examples for helping you to know how to find the logarithm of any quantity on the basis of another quantity easily in mathematics.

$(1) \,\,\,$ Evaluate $\log_{2}{128}$

Express $128$ as factors in terms of $2$.

$\implies$ $128$ $\,=\,$ $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

Count the total factors of $2$ in the product.

$\implies$ $128$ $\,=\,$ $\underbrace{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}_{7}$

Write the value of log of $128$ to the base $2$.

$\,\,\, \therefore \,\,\,\,\,\, \log_{2}{128} \,=\, 7$

The total number of factors is $7$ when the quantity $128$ is written as factors on the basis of $2$.

$(2) \,\,\,$ Find $\log_{9}{81}$

Express the quantity $81$ as factors in terms of $9$.

$\implies 81 \,=\, 9 \times 9$

Count the total number of factors of $9$ in the product.

$\implies 81 \,=\, \underbrace{9 \times 9}_{2}$

Write the value of logarithm of $81$ to the base $9$.

$\,\,\, \therefore \,\,\,\,\,\, \log_{9}{81} \,=\, 2$

The number of factors is $2$ if the number $81$ is written as factors on the basis of $9$.

$(3) \,\,\,$ Evaluate $\log_{5}{125}$

Write the quantity $125$ as factors in terms of $5$.

$\implies 125 \,=\, 5 \times 5 \times 5$

Count the total factors of $5$ in the product.

$\implies 125 \,=\, \underbrace{5 \times 5 \times 5}_{3}$

Write the value of logarithm of $125$ to the base $5$.

$\,\,\, \therefore \,\,\,\,\,\, \log_{5}{125} \,=\, 3$

The total number of factors is $3$ if the number $125$ is expressed as factors on the basis of $5$.

$(4) \,\,\,$ Calculate $\log_{10}{10000}$

Write the quantity $10000$ as factors in terms of $10$.

$\implies 10000 \,=\, 10 \times 10 \times 10 \times 10$

Count the total number of factors of $10$ in this product.

$\implies 10000 \,=\, \underbrace{10 \times 10 \times 10 \times 10}_{4}$

Write the value of logarithm of $10000$ to the base $10$.

$\,\,\, \therefore \,\,\,\,\,\, \log_{10}{10000} \,=\, 4$

Therefore, the total number of factors is $4$ if the quantity $10000$ is written as factors on the basis of $10$.

$(5) \,\,\,$ Calculate $\log_{4}{1024}$

Express the quantity $1024$ as factors in terms of $4$.

$\implies 1024$ $\,=\,$ $4 \times 4 \times 4 \times 4 \times 4$

Count the total number of factors of $4$ in this product.

$\implies 1024$ $\,=\,$ $\underbrace{4 \times 4 \times 4 \times 4 \times 4}_{5}$

Write the value of log of $1024$ to the base $4$.

$\,\,\, \therefore \,\,\,\,\,\, \log_{4}{1024} \,=\, 5$

So, the total number of factors is $5$ when the quantity $1024$ is written as factors on the basis of $4$.

$(6) \,\,\,$ Evaluate $\log_{0.3}{0.027}$

Write the quantity $0.027$ as factors in terms of $0.3$.

$\implies 0.027 \,=\, 0.3 \times 0.3 \times 0.3$

Now, count the total factors of $0.3$ in the product.

$\implies 0.027 \,=\, \underbrace{0.3 \times 0.3 \times 0.3}_{3}$

Write the value of logarithm of $0.027$ to the base $0.3$.

$\,\,\, \therefore \,\,\,\,\,\, \log_{0.027}{0.3} \,=\, 3$

Therefore, the total number of factors is $3$ if $0.027$ is expressed as factors on the basis of $0.3$.

$(7) \,\,\,$ Evaluate $\log_{\small \dfrac{1}{8}}{\Bigg(\dfrac{1}{512}\Bigg)}$

The quantity and base quantity both are in fraction from. The quantity $512$ can be written in terms of $8$. So, write the quantity $\dfrac{1}{512}$ as factors in terms of $\dfrac{1}{8}$.

$\implies \dfrac{1}{512} \,=\, \dfrac{1}{8} \times \dfrac{1}{8} \times \dfrac{1}{8}$

Now, count the total factors of $\dfrac{1}{8}$ in the product.

$\implies \dfrac{1}{512} \,=\, \underbrace{\dfrac{1}{8} \times \dfrac{1}{8} \times \dfrac{1}{8}}_{3}$

Express the value of logarithm of $\dfrac{1}{512}$ to the base $\dfrac{1}{8}$.

$\,\,\, \therefore \,\,\,\,\,\, \log_{\tiny \dfrac{1}{8}}{\Bigg(\dfrac{1}{512}\Bigg)} \,=\, 3$

So, the total number of factors is $3$ if the fraction $\dfrac{1}{512}$ is written as factors on the basis of another fraction $\dfrac{1}{8}$.

$(8) \,\,\,$ Find $\log_{\sqrt{3}}{9}$

It’s a special case. In this example, the quantity is a whole number but the base of logarithm is an irrational number. However, the logarithm of $9$ to base $\sqrt{3}$ can be evaluated.

Express the quantity $9$ as factors in terms of $\sqrt{3}$.

$\implies 9$ $\,=\,$ $3 \times 3$

$\implies 9$ $\,=\,$ $\sqrt{3} \times \sqrt{3} \times \sqrt{3} \times \sqrt{3}$

Count the total number of factors of $\sqrt{3}$ in this product.

$\implies 0$ $\,=\,$ $\underbrace{\sqrt{3} \times \sqrt{3} \times \sqrt{3} \times \sqrt{3}}_{4}$

Write the value of log of $9$ to the base $\sqrt{3}$.

$\,\,\, \therefore \,\,\,\,\,\, \log_{\sqrt{3}}{9} \,=\, 4$

Therefore, the total number of factors is $4$ when the number $9$ is written as factors on the basis of $\sqrt{3}$.

$(9) \,\,\,$ Find $\log_{a}{a^3}$

It is another special case. The quantity and base quantity are in algebraic form. However, repeat the same procedure to find the logarithm of $a^3$ to base $a$.

Express the quantity $a^3$ as factors in terms of $a$.

$\implies a^3$ $\,=\,$ $a \times a \times a$

Count total number of factors of $a$ in this product.

$\implies a^3$ $\,=\,$ $\underbrace{a \times a \times a}_{3}$

Write the value of log of $a^3$ to the base $a$.

$\,\,\, \therefore \,\,\,\,\,\, \log_{a}{a^3} \,=\, 3$

Therefore, the total number of factors is $3$ when the quantity $a^3$ is written as factors on the basis of $a$.

$(10) \,\,\,$ Evaluate $\log_{2}{11}$

Write the number $11$ as factors in terms of $2$ but it is not possible to write the quantity $11$ as factors in terms of $2$ but don’t think it is impossible. Actually, it’s not possible for you at this time if you are newly learning logarithms. So, let us continue learning the logarithms to advanced level.

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