Question

Find the numerical value of the log expression $\log \dfrac{{{a}^{4}}{{b}^{6}}}{{{c}^{8}}}$ if $\log a=-5$, $\log b=-7$ and $\log c=1$.

Answer 1

To obtain the solution of the given log expression we will use the properties of logarithm function. Firstly we will separate the numerator and denominator value of $\log \dfrac{{{a}^{4}}{{b}^{6}}}{{{c}^{8}}}$ in such way that we can use the other values given in it. Then we will solve it to get the desired answer.

Complete step-by-step solution:
To find the numerical value of the log expression:
$\log \dfrac{{{a}^{4}}{{b}^{6}}}{{{c}^{8}}}$……$\left( 1 \right)$
The values given are:
$\log a=-5$
$\log b=-7$
$\log c=1$
Now, we will use the logarithm identity given below in equation (1)
$\log \dfrac{a}{b}=\log a-\log b$
So we get,
$\log \dfrac{{{a}^{4}}{{b}^{6}}}{{{c}^{8}}}=\log {{a}^{4}}{{b}^{6}}-\log {{c}^{8}}$……$\left( 3 \right)$
Next, we will use the logarithm identity given below in equation (3):
$\log ab=\log a+\log b$
So we get,
$\log \dfrac{{{a}^{4}}{{b}^{6}}}{{{c}^{8}}}=\log {{a}^{4}}+\log {{b}^{6}}-\log {{c}^{8}}$…….$\left( 4 \right)$
Now, we will use the below identity in equation (4):
$\log {{a}^{b}}=b\log a$
So we get,
$\log \dfrac{{{a}^{4}}{{b}^{6}}}{{{c}^{8}}}=4\log a+6\log b-8\log c$……$\left( 5 \right)$
The values given are:
$\log a=-5$
$\log b=-7$
$\log c=1$
Substituting above value in equation (5) and simplify,
$\begin{aligned} & \Rightarrow \log \dfrac{{{a}^{4}}{{b}^{6}}}{{{c}^{8}}}=4\times \left( -5 \right)+6\times \left( -7 \right)-8\times 1 \\\ & \Rightarrow \log \dfrac{{{a}^{4}}{{b}^{6}}}{{{c}^{8}}}=-20-42-8 \\\ & \therefore \log \dfrac{{{a}^{4}}{{b}^{6}}}{{{c}^{8}}}=-70 \\\ \end{aligned}$
Hence the numerical value of $\log \dfrac{{{a}^{4}}{{b}^{6}}}{{{c}^{8}}}$ is $-70$

Note: The logarithm function used is inverse function of the exponential function. It is generally defined as $y={{\log }_{b}}x$ where $b$ is the base and it is read as “log base $b$ of $x$”. The logarithm with base 10 is known as common logarithm and is generally used when the base is not stated. The identities of logarithm functions are of great use for calculating higher problems involving big numbers as the product term can be changed to addition and we all know that addition of big numbers is easier than the product of big numbers. The only thing that is necessary to use the identities is that the base should be the same.