Solve each equation for the indicated variable. Use logarithms with the appropriate bases. log A = log B - C log x, for A

Verified step by step guidance

1. Start with the given equation: \(\log A = \log B - C \log x\).
2. Recall the logarithm property that allows combining terms: \(a \log b = \log b^a\). Use this to rewrite \(C \log x\) as \(\log x^C\), so the equation becomes \(\log A = \log B - \log x^C\).
3. Use the logarithm subtraction rule: \(\log m - \log n = \log \left( \frac{m}{n} \right)\), to combine the right side into a single logarithm: \(\log A = \log \left( \frac{B}{x^C} \right)\).
4. Since \(\log A = \log \left( \frac{B}{x^C} \right)\), the arguments must be equal (assuming the logs have the same base), so set \(A = \frac{B}{x^C}\).
5. Solve for \(x\) by isolating it: multiply both sides by \(x^C\) to get \(A x^C = B\), then divide both sides by \(A\) to get \(x^C = \frac{B}{A}\). Finally, take the \(C\)-th root of both sides to solve for \(x\): \(x = \left( \frac{B}{A} \right)^{\frac{1}{C}}\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential. These properties allow you to combine or separate logarithmic expressions, making it easier to isolate variables and simplify equations.

Solving Logarithmic Equations

Solving logarithmic equations involves rewriting the equation to isolate the logarithmic term and then converting it to its exponential form. This process helps in solving for the variable inside the logarithm by removing the log function.

Change of Base and Logarithm Bases

Recognizing the base of logarithms and using the change of base formula when necessary is important. This allows you to work with logarithms of different bases and apply appropriate logarithmic rules to solve for the variable.