Question

Simplify $\log \left( {1.77} \right)$.

Answer 1

Hint: Here, you are asked to simplify the given logarithmic equation - $\log \left( {1.77} \right)$. In order to simplify this logarithm, you need to consider the properties which a logarithmic quantity possesses. Try to use these properties and make arrangements appropriately and simplify it.

Complete step by step answer:
The given number is $\log \left( {1.77} \right)$. Let us look for some properties of logarithm.
If you are given two numbers $a\& b$, then the logarithm of the product of the number $a$ and $b$ is equal to the sum of individual logarithms of the numbers. Mathematically,
$\log \left( {ab} \right) = \log a + \log b$
If you are given two numbers $a\& b$, then the logarithm of $a$ divided by $b$ is equal to the logarithm of $a$ minus logarithm of $b$. Mathematically,
$\log \left( {\dfrac{a}{b}} \right) = \log a - \log b$
Logarithm of $a$ to the base $b$ is ${\log _b}a$. If ${\log _b}a = c$, then ${b^c} = a$.

Let us find some logarithms. The Logarithm of one is zero, that is ${\log _b}1 = 0$, because anything raised to zero is one and $b$ is any real number. One more property is,
${\log _b}{a^c} = c{\log _b}a$
Let us consider the given number $\log \left( {1.77} \right)$.
$1.77$ can be written as $\dfrac{{177}}{{100}}$.
$\log \left( {1.77} \right) = \log \left( {\dfrac{{177}}{{100}}} \right)$, now, let us use the property which gives the logarithm if a certain number is divided by some number.

Therefore, $\log \left( {\dfrac{{177}}{{100}}} \right) = \log 177 - \log 100$. When no base is mentioned, the default base is $10$ and therefore, $\log 177 - \log 100 = {\log _{10}} - {\log _{10}}100$. As you know that square of ten is equal to hundred, ${10^2} = 100$, we can use the power property discussed above, so that
\[
{10^2} = 100 \to {b^c} = a \\
\Rightarrow {\log _{10}}100 = 2 \\ \]
$\Rightarrow\log \left( {1.77} \right) = \log \left( {177} \right) - 2$
Factors of $177$ are $1,3,59\& 177$. $177$ can be written as $3 \times 59$. Let substitute this in the equation,
$\log \left( {177} \right) = \log \left( {3 \times 59} \right)$
Now, use the logarithmic property of products. We get,
$\log \left( {3 \times 59} \right) = \log 3 + \log 59$
Finally, we get $\log \left( {1.77} \right) = \log 3 + \log 59 - 2$.

Hence, $\log \left( {1.77} \right)$ is simplified to \[\log 3 + \log 59 - 2\].

Note: The whole procedure you are supposed to understand carefully. You need to learn how to break down numbers so you can simplify the given problem. Also, keep in mind the various properties of logarithm such as logarithm of products of two numbers, logarithm of a rational number and power property.