Question

How do you solve $\ln \left( {x - 1} \right) = 5?$

Answer 1

The given question involves the operation of addition/ subtraction/ multiplication/ division. We need to know the converting process of natural logarithmic function into exponential function. We need to find the value $x$ from the given equation. Also, we need to know how to calculate ${e^x}$ in the scientific calculator. The final answer would be a decimal number or whole number.

Complete step by step solution:
The given equation in the question is shown below,
$\ln \left( {x - 1} \right) = 5 \to \left( 1 \right)$
For solving the above equation take exponent on both sides of the equation$\left( 1 \right)$,
we get

$$\left( 1 \right) \to \ln \left( {x - 1} \right) = 5 \\\ {e^{\ln \left( {x - 1} \right)}} = {e^5} \to \left( 2 \right) \\\$$

We know that,
${e^{\ln x}} = x$
So, by using the above equation we can modify the equation $\left( 2 \right)$, as follows

$$\left( 2 \right) \to {e^{\ln \left( {x - 1} \right)}} = {e^5} \\\ x - 1 = {e^5} \\\$$

Let’s separate the $x$ terms and constant terms in the above equation, we get
$x = 1 + {e^5} \to \left( 3 \right)$
By using a calculator, we have
${e^5} = 148.413$
So, the equation$\left( 3 \right)$becomes,

$$\left( 3 \right) \to x = 1 + {e^5} \\\ x = 1 + 148.413 \\\$$

$x = 149.413$
So, the final answer is,
$x = 149.413$

Note: This type of question involves the operation of addition/ subtraction/ multiplication/ division. Note that the exponent term is the inverse function of a natural logarithmic function. So, when exponent and natural logarithm are involved in a single term, the final answer would be $1$. Note that ${e^x}$ terms can easily be calculated with the help of a scientific calculator. Note that to solve this type of question we would find the value of $x$ from the given equation. Note that anything power zero will be $1$ and zero power anything becomes zero. Remember the basic exponent and natural logarithmic conditions to solve these types of questions. Also, note that every answer should have at least three numbers after the decimal point in a decimal number.