Solve \(\ln (5 x+1)=3\)

Round to the nearest thousandth.

Exponentiate both sides of the equation with base e : e l n ( 5 x + 1 ) = e 3 .
Simplify the equation: 5 x + 1 = e 3 .
Isolate x : x = 5 e 3 − 1 ​ .
Calculate and round to the nearest thousandth: x ≈ 3.817 ​ .

Explanation

Problem Analysis We are given the equation ln ( 5 x + 1 ) = 3 and asked to solve for x , rounding to the nearest thousandth.

Exponentiating Both Sides To solve for x , we first exponentiate both sides of the equation with base e to remove the natural logarithm. This gives us: e l n ( 5 x + 1 ) = e 3

Simplifying the Equation Since e l n ( u ) = u , we can simplify the left side of the equation to get: 5 x + 1 = e 3

Isolating the Term with x Now, we subtract 1 from both sides of the equation: 5 x = e 3 − 1

Solving for x Next, we divide both sides by 5 to isolate x :
x = 5 e 3 − 1 ​

Calculating the Value of x We know that e 3 ≈ 20.085536923187668 , so we can substitute this value into the equation: x = 5 20.085536923187668 − 1 ​ = 5 19.085536923187668 ​ ≈ 3.8171073846375334

Rounding to the Nearest Thousandth Finally, we round the value of x to the nearest thousandth: x ≈ 3.817


Examples
Solving logarithmic equations is useful in various fields such as finance, physics, and engineering. For example, in finance, it can be used to calculate the time it takes for an investment to reach a certain value with compound interest. In physics, it can be used to model radioactive decay. Understanding how to solve these equations allows us to make predictions and analyze real-world phenomena.

Answered by GinnyAnswer | 2025-07-08