Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. 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. To isolate \( \log A \), add \( C \log x \) to both sides: \( \log A + C \log x = \log B \).
3. Factor out \( \log x \) from the left side: \( \log A = \log B - C \log x \) becomes \( \log A + C \log x = \log B \).
4. Rearrange the equation to solve for \( A \): \( \log A = \log B - C \log x \).
5. Exponentiate both sides to remove the logarithm and solve for \( A \): \( A = 10^{(\log B - C \log x)} \).

Key Concepts

Logarithmic Properties

Understanding the properties of logarithms is essential for manipulating logarithmic equations. Key properties include the product, quotient, and power rules, which allow us to combine or separate logarithmic expressions. For instance, log(A) - log(B) can be rewritten as log(A/B), facilitating the solving process.

Isolating Variables

Isolating a variable involves rearranging an equation to solve for a specific variable. This often requires using inverse operations, such as exponentiation for logarithmic equations. In the given equation, isolating 'x' means expressing it in terms of 'A', 'B', and 'C' by systematically applying logarithmic properties.

Change of Base Formula

The change of base formula is crucial when dealing with logarithms of different bases. It states that log_b(a) can be converted to log_k(a) / log_k(b) for any positive base k. This is particularly useful when the equation requires logarithms in a specific base that differs from the one given, allowing for easier calculations.