Question

How do you write the equivalent logarithmic equation \[{e^{ - x}} = 5\] ?

Answer 1

Hint: The given equation is in the form of exponential and as we know that the exponential function and the logarithmic function are inverse of each other. Here apply logarithm on both sides of the equation and we simplify the equation and hence we determine the solution for the given question.

Complete step-by-step answer:
The logarithmic function and the exponential function are both inverse of each other. The exponential number can be written in the form of a logarithmic number and likewise we can write the logarithmic number in the form of an exponential number.
Now consider the given equation
 \[{e^{ - x}} = 5\]
Apply the logarithm on both sides we get
 \[ \Rightarrow \log {e^{ - x}} = \log 5\]
The logarithm function and the exponential function are inverse to each other it will gets cancels and the above equation is written as
 \[ \Rightarrow - x = \log 5\]
Multiply the above equation by -1 and we get
 \[ \Rightarrow x = - \log 5\]
Hence the logarithmic equation for \[{e^{ - x}} = 5\] is \[x = - \log 5\]
Hence we have determined the solution for the given equation
We can also find the value of log 5 by using the Clark’s table or we write it as it is.
So, the correct answer is “ \[x = - \log 5\] ”.

Note: The exponential number is inverse of logarithmic. But here we have not used this. We have applied the log on both terms. The logarithmic functions have several properties on addition, subtraction, multiplication, division and exponent. So we have to use logarithmic properties. We have exact values for the numerals by the Clark’s table, with the help of it we can find the exact value.