Question

Express y as a function of \(x .\) The constant \(C\) is a positive number. \(\ln (y+4)=5 x+\ln C\)

Short answer

y = C \times e^{5x} - 4

Step-by-step solution

Isolate the logarithmic expression

Given the equation \(\text{ln}(y+4) = 5x + \text{ln}C\), let's focus on isolating the term with y. The equation is already isolated as \(\text{ln}(y+4)\) on the left-hand side.

Exponentiate both sides to eliminate the natural logarithm

Since the natural logarithm function is the inverse of the exponential function, we'll exponentiate both sides of the equation to remove the logarithm: \[ e^{\text{ln}(y+4)} = e^{5x + \text{ln}C} \].

Simplify the exponential expression

Using the property of exponentiation, where \( e^{a+b} = e^a \times e^b \), the right-hand side becomes: \[ e^{5x + \text{ln}C} = e^{5x} \times e^{\text{ln}C} = e^{5x} \times C \]. Therefore, the simplified equation is:\[ y + 4 = C \times e^{5x} \].

Solve for y

To isolate y, subtract 4 from both sides of the equation: \[ y = C \times e^{5x} - 4 \].

Key concepts

Natural Logarithm

The natural logarithm, denoted as \(\text{ln}\), is the logarithm to the base e, where e is an irrational constant approximately equal to 2.71828. It's a fundamental concept in calculus and is used to solve equations involving continuous growth or decay.
When working with natural logarithms, the key property to remember is that \(\text{ln}(e) = 1\). This property can help simplify complex expressions involving e and natural logarithms.

Exponentiation

Exponentiation is the process of raising a number, known as the base, to a power. It is the inverse operation of the logarithm. For instance, if \(\text{ln}(y+4) = 5x + \text{ln}C\), and we exponentiate both sides of the equation, we use the property that \(\text{ln}(a) = b\) translates to \(\text{e}^b = a\).
This property helps to get rid of the logarithm and bring the equation back to a form that is easier to manipulate and solve.

Solving Equations

Solving equations involving logarithms and exponentiation often requires isolating variables and using properties of logarithms and exponents to simplify the terms. For example, to solve \(\text{ln}(y+4)=5x+\text{ln}C\), we exponentiate both sides to eliminate the natural logarithm, resulting in \[ e^{\text{ln}(y+4)} = e^{5x + \text{ln}C} \].
After exponentiation, the equation becomes manageable, allowing us to isolate the variable y step-by-step.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate the variable of interest. This often includes applying arithmetic operations such as addition, subtraction, multiplication, and division.
In the example, after exponentiating and simplifying: \[ y + 4 = C \times e^{5x} \]
We then isolate y by subtracting 4 from both sides: \[ y = C \times e^{5x} - 4 \].
These steps demonstrate the power of algebraic manipulation in expressing one variable as a function of another.