Question

The value of $\log 4+\dfrac{1}{3}\log 125-\dfrac{1}{5}\log 32$ is equal to
A. 7
B. 3
C. 1
D. 5

Answer 1

Hint: We solve this question by using the basic functions and formulae for the logarithms. We know that $a\log b=\log {{b}^{a}}.$ Using this, we simplify the individual terms. Then we use the formulae for addition and subtraction of log terms and obtain the value.

Complete step by step answer:
In order to solve this question, let us first write down the given expression.
$\Rightarrow \log 4+\dfrac{1}{3}\log 125-\dfrac{1}{5}\log 32$
We can represent the terms 125 as a power of 5 as ${{5}^{3}}$ and 32 as a power of 2 given by ${{2}^{5}}.$
$\Rightarrow \log 4+\dfrac{1}{3}\log {{5}^{3}}-\dfrac{1}{5}\log {{2}^{5}}$
We know that $a\log b=\log {{b}^{a}}.$ Using this for the second and third terms,
$\Rightarrow \log 4+\log {{5}^{3\times \left( \dfrac{1}{3} \right)}}-\log {{2}^{5\times \left( \dfrac{1}{5} \right)}}$
Cancelling the terms in the powers since they have the same numerators and denominators,
$\Rightarrow \log 4+\log 5-\log 2$
Now this is just the simple addition and subtraction of log terms. This is done using the formula for addition of log terms as $\log a+\log b=\log \left( a.b \right).$ Using this for the first two terms in the above equation,
$\Rightarrow \log \left( 4\times 5 \right)-\log 2$
Multiplying the two terms,
$\Rightarrow \log 20-\log 2$
The formula for subtraction of two log terms is given as $\log a-\log b=\log \left( \dfrac{a}{b} \right).$ Using this for the above equation,
$\Rightarrow \log \left( \dfrac{20}{2} \right)$
Dividing the two terms,
$\Rightarrow \log \left( 10 \right)$
We know the term log is always to the base 10. We can convert from the log term to an exponential form such that if ${{\log }_{10}}a=b,$ ${{10}^{b}}=a.$ Using this, if
$\Rightarrow {{\log }_{10}}10=x$
Then,
$\Rightarrow {{10}^{x}}=10$
Equating the two powers,
$\Rightarrow x=1$

So, the correct answer is “Option C”.

Note: We need to know the concept of logarithms and their basic operations to solve such questions. It is important to note the formula for converting from the log form to exponential form. Log terms are always to the base 10 unless specified and ln terms are always to the base e unless specified.