Question

Find the value of x if the expression ${\log _{10}}\left( {x + 5} \right) = 1$.
$
  (a){\text{ 3}} \\
  (b){\text{ 4}} \\
  (c){\text{ 5}} \\
  (d){\text{ 6}} \\
 $

Answer 1

Hint – In this problem use the property of logarithm that ${\log _a}b = \dfrac{{\log b}}{{\log a}}$ in the left hand side of the given expression, then compare log values both sides to get the required value of x.

Complete step-by-step answer:
Given expression
${\log _{10}}\left( {x + 5} \right) = 1$
As we know that ${\log _a}b = \dfrac{{\log b}}{{\log a}}$ so use this property of logarithmic in above equation we have,
$ \Rightarrow \dfrac{{\log \left( {x + 5} \right)}}{{\log 10}} = 1$
$ \Rightarrow \log \left( {x + 5} \right) = \log 10$
Comparison of log on both sides we have,
$ \Rightarrow x + 5 = 10$
$ \Rightarrow x = 10 - 5 = 5$
So this is the required value of x.
Hence option (C) is correct.

Note – Such problems are simply based upon logarithmic properties. Some of the important logarithm properties include ${\log _b}{a^n} = n{\log _b}a$, ${\log _b}a = p \Rightarrow a = {b^p}$ and some are being mentioned above. Logarithmic functions are basically the inverse of exponential functions.