Question

How do you solve $ \log x = 4 $ ?

Answer 1

Hint: In this question we need to solve $ \log x = 4 $ . Here, we have the bases of $ \log $ as default is $ e $ and $ 10 $ . Therefore, we will apply both the bases separately and determine the value of $ x $ respectively. By using the rule that we already know from the definition of logarithm as $ {\log _a}b = c $ $ \Leftrightarrow $ $ {a^c} = b $ .

Complete step-by-step answer:
Here, we need to solve $ \log x = 4 $ .
We know two bases of the $ \log $ as default i.e., $ e $ and $ 10 $ .
First let us consider the base of the $ \log $ as $ 10 $ ,
 $ {\log _{10}}x = 4 $
As we know the log form and exponential form are interchangeable, we have,
 $ {\log _a}b = c $
This can be written as,
 $ {a^c} = b $
Therefore, by using this let us rewrite the equation $ {\log _{10}}x = 4 $ as,
  $ x = {10^4} $
Hence, $ x = 10000 $
So, the correct answer is “ $ x = 10000 $ ”.

Next, let us consider the base of the $ \log $ as $ e $ .
 $ {\log _e}x = 4 $
Therefore, let us rewrite the equation $ {\log _e}x = 4 $ as,
 $ x = {e^4} $
Now, we know that approximately the value of $ e = 2.718 $
Now, let us apply the value,
 $ x = {\left( {2.718} \right)^4} $
Hence, $ x = 54.57 $
So, the correct answer is “ $ x = 54.57 $ ”.

Note: In this question it is important to note here that considering the base of the log as $ 10 $ is the most common method used for solving these types of questions. We consider $ e $ as the base because exponential form is the inverse of logarithm. Logarithms are the opposite of exponentials, just as subtraction is the opposite of addition and multiplication i.e., a logarithm says how many of one number to multiply to get another number and the exponent of a number says how many times to use the number in a multiplication. And, from the definition of logarithm, if $ a $ and $ b $ are positive real numbers and $ a \ne 1 $ , then $ {\log _e}x = 4 $ is equivalent to $ {a^c} = b $ . If we can remember this relation, then we will not have too much trouble with logarithms.