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.