Question

If ${\log _{10}}2 = 0.3010$, then the value of ${\log _{10}}80$ is
A.$1.6020$
B.$1.9030$
C.$3.9030$
D.None of these

Answer 1

Hint Here, we will use the logarithm property, ${\log _b}ac = {\log _b}a + {\log _b}c$and then the power rule of logarithm, ${\log _b}\left( {{a^c}} \right) = c{\log _b}a$. Then we will use the property of logarithm, ${\log _b}b = 1$ accordingly in the given expression to simplify it.

Complete step-by-step answer:
We are given that the ${\log _{10}}2 = 0.3010$.
We will now rewrite the expression ${\log _{10}}80$, we get
$\Rightarrow {\log _{10}}\left( {8 \times 10} \right)$
Using the logarithm property, ${\log _b}ac = {\log _b}a + {\log _b}c$ in the above expression, we get

$$\Rightarrow {\log _{10}}8 + {\log _{10}}10 \\\ \Rightarrow {\log _{10}}\left( {{2^3}} \right) + {\log _{10}}10 \\\$$

Let us now make use of the power rule of logarithm, ${\log _b}\left( {{a^c}} \right) = c{\log _b}a$.
So, on applying this rule in the above equation, we get
$\Rightarrow 3{\log _{10}}2 + {\log _{10}}10$
Using the property of logarithm, ${\log _b}b = 1$ in the above equation, we get
$\Rightarrow 3{\log _{10}}2 + 1$
Substituting the value of ${\log _{10}}2$ in the above expression, we get

$$\Rightarrow 3 \times 0.3010 + 1 \\\ \Rightarrow 0.9030 + 1 \\\ \Rightarrow 1.9030 \\\$$

Thus, the value of ${\log _{10}}80$ is $1.9030$.
Hence, option B is correct.

Note The power rule can be used for fast exponent calculation using multiplication operation. Students should make use of the appropriate formula of logarithms wherever needed and solve the problem. In mathematics, if the base value in the logarithm function is not written, then the base is $e$.