Solve each equation for the indicated variable. Use logarithms with the appropriate bases. p = a + (k/ln x), for x

Verified step by step guidance

1. Start with the given equation: \(p = a + \frac{k}{\ln x}\).
2. Isolate the term containing \(x\) by subtracting \(a\) from both sides: \(p - a = \frac{k}{\ln x}\).
3. Next, solve for \(\ln x\) by taking the reciprocal and multiplying both sides by \(k\): \(\ln x = \frac{k}{p - a}\).
4. Recall that \(\ln x\) means the natural logarithm of \(x\), so to solve for \(x\), rewrite the equation in exponential form: \(x = e^{\frac{k}{p - a}}\).
5. This expression gives \(x\) in terms of \(p\), \(a\), and \(k\), using the natural exponential function.

Key Concepts

Solving Equations for a Specific Variable

This involves isolating the variable of interest on one side of the equation. It requires algebraic manipulation such as addition, subtraction, multiplication, division, and applying inverse operations to rewrite the equation in terms of the desired variable.

Properties of Logarithms

Logarithms are the inverses of exponential functions and have specific properties like the product, quotient, and power rules. Understanding these properties helps simplify expressions and solve equations involving logarithms with different bases.

Natural Logarithm and the Constant e

The natural logarithm (ln) is the logarithm with base e, where e ≈ 2.718. It is commonly used in calculus and algebra to solve equations involving growth, decay, or continuous compounding. Recognizing when to apply ln and how to manipulate it is essential for solving equations like p = a + (k/ln x).