Solve for $x$.
$\ln (x+3)-\ln 4=4$
Do not round any intermediate computations, and round your answer to the nearest hundred
$x=\square$
Answer (1)
Use the logarithm property ln a − ln b = ln b a to rewrite the equation as ln 4 x + 3 = 4 .
Exponentiate both sides of the equation with base e to get 4 x + 3 = e 4 .
Multiply both sides by 4 to get x + 3 = 4 e 4 .
Subtract 3 from both sides to get x = 4 e 4 − 3 ≈ 215.39 .
215.39
Explanation
Problem Analysis We are given the equation ln ( x + 3 ) − ln 4 = 4 and we want to solve for x . We will use properties of logarithms to isolate x .
Applying Logarithm Properties Using the logarithm property ln a − ln b = ln b a , we can rewrite the equation as ln 4 x + 3 = 4 .
Exponentiating Both Sides To remove the natural logarithm, we exponentiate both sides of the equation with base e . This gives us e l n 4 x + 3 = e 4 , which simplifies to 4 x + 3 = e 4 .
Isolating x Next, we multiply both sides of the equation by 4 to isolate the term with x . This gives us x + 3 = 4 e 4 .
Solving for x Finally, we subtract 3 from both sides to solve for x : x = 4 e 4 − 3 .
Calculating the Value and Rounding Now, we need to calculate the value of 4 e 4 − 3 and round the result to the nearest hundred. Using a calculator, we find that 4 e 4 − 3 ≈ 4 ( 54.598 ) − 3 ≈ 218.392 − 3 ≈ 215.392 . Rounding to the nearest hundred, we get 215.39 .
Examples
Logarithmic equations are used in various fields such as finance, physics, and engineering. For example, in finance, they can be used to model the growth of investments or the decay of loans. In physics, they can be used to describe the behavior of sound waves or the decay of radioactive materials. Understanding how to solve logarithmic equations is essential for making informed decisions and solving real-world problems in these fields. For instance, calculating the time it takes for an investment to double at a certain interest rate involves solving a logarithmic equation. This skill is crucial for financial planning and investment analysis.
