Question

The value of logarithm $\log 5 - 1$ can be written as a single logarithm with base as 10 is ______
A). $\log \dfrac{1}{2}$
B). $\log 1$
C). $\log 5$
D). $\log 2$

Answer 1

To solve this question first we assume a variable equal to the given expression. Then first we write 1 in a logarithm function with the base 10. After that, we simplify that using the property of difference of two logarithmic functions. After doing that and simplifying we get that whole expression in a single logarithm with a base equal to 10.

Complete step-by-step solution:
Given,
The base of the logarithm is 10
To find,
The value of $\log 5 - 1$
Let, $x = \log 5 - 1$ ……(i)
Now using the rule of logarithm ${\log _a}^a = 1$
${\log _{10}}$ is given so here the value of $a$ is 10
On putting these value in the logarithm property
${\log _{10}}{10} = 1$
On putting the value in equation 1
$x = \log 5 - \log 10$
On using the property of logarithm $\log \dfrac{a}{b} = \log a - \log b$
$x = \log \dfrac{5}{{10}}$
On further solving we get the value as
$x = \log \dfrac{1}{2}$
Final answer:
The value of the expression $\log 5 - 1$ in a single logarithm with base equal to 10
$\log 5 - 1 = \log \dfrac{1}{2}$
According to the obtained answer option, A is the correct answer.

Note: In this question 1 is outside the logarithm function. Students commit mistakes by taking that in a logarithm function if the answers are not given and if the answer is matched then tick the incorrect answer. If the base of the logarithm function is not given then we assume that base equal to 10 is by default. And if the logarithm function has a base equal to $e$ then that function is written as $\ln$. To solve this type of question you must know the all properties of the logarithm.