Question

How do you calculate the logarithmic function ${{\log }_{10}}345$ ?

Answer 1

We have been given a logarithmic expression with base equal to 10 and a constant argument equal to 345. We shall use various properties of the functions in logarithm to simplify and calculate the given expression. We will break 345 as the product of multiple numbers which would be then expressed as simpler logarithmic functions whose values are known by us. After substituting the values of these logarithmic functions, we shall compute the final answer.

Complete step by step solution:
Given that ${{\log }_{10}}345$.
In order to simplify this expression, we shall use the basic properties of logarithmic functions.
We know that $345=3\times 5\times 23$.
$\Rightarrow {{\log }_{10}}345={{\log }_{10}}\left( 3 \right)\left( 5 \right)\left( 23 \right)$
According to a property of logarithm functions, when a logarithm function is expressed in terms of the product of n terms, then that logarithm function can be expressed as the sum of n individual logarithmic functions of those n terms, that is, $\log ab=\log a+\log b$.
Using this property on the given expression, we get
$\Rightarrow {{\log }_{10}}345={{\log }_{10}}3+{{\log }_{10}}5+{{\log }_{10}}23$
Now, we shall substitute the values of these three individual logarithmic expressions which have a base 10.
We know that ${{\log }_{10}}3=0.4771,{{\log }_{10}}5=0.6989$ and ${{\log }_{10}}23=1.3617$. Putting these values in our expression, we get
$\Rightarrow {{\log }_{10}}345=0.4771+0.6989+\operatorname{l}.3617$
$\Rightarrow {{\log }_{10}}345=2.5377$
Therefore, ${{\log }_{10}}345$ is equal to 2.5377.

Note: Another method of solving this problem was by taking this expression equal to some variable-x. Then, we would take the antilog of the entire expression modifying it as $345={{10}^{x}}$. Then, we would perform calculations with the help of a logarithmic calculator to find the value of this variable-x which would be equal to our final answer.