91Ó°ÊÓ

Logarithms help us to solve for unknown exponents and manage multiplication and division within logarithmic expressions. Some essential properties include:
- **Product Rule**: \(\text{log}_b(a \times c) = \text{log}_b(a) + \text{log}_b(c)\). This enables us to compress the sum of two logs into a single logarithm.
- **Quotient Rule**: \(\text{log}_b \left( \frac{a}{c} \right) = \text{log}_b(a) - \text{log}_b(c)\). This is handy for combining differences into a single logarithm.
- **Power Rule**: \(\text{log}_b(a^c) = c \times \text{log}_b(a)\). This brings the exponent out as a coefficient.
Understanding these properties is fundamental. For instance, \(2 \text{log}_3(x) = \text{log}_3(64)\) can be solved with these rules. We rewrite \(2 \text{log}_3(x)\) as \(\text{log}_3(x^2)\), simplifying to \(\text{log}_3(64)\). Then, \(x^2 = 64\) leads us to \(x = 8\).

Solving logarithm equations involves using logarithm properties to isolate the variable. Here is the general process:
1. **Apply Logarithm Properties**: Simplify the given logarithmic expressions using product, quotient, or power rules.
2. **Isolate the Logarithm**: Rearrange the equation to isolate the logarithmic expression on one side.
3. **Convert to Exponential Form**: If needed, rewrite the logarithmic equation in its exponential form to solve for the variable.
Take the equation \(\frac{3}{2} \text{log}_7(x) = \text{log}_7(125)\). We isolate \(\text{log}_7(x)\) by multiplying both sides by \(\frac{2}{3}\). This transforms the equation into \(\text{log}_7(x) = \text{log}_7(25)\). Since the log bases and values are equal, it follows that \(x = 25\).

Logarithm rules allow us to manipulate logarithmic expressions effectively. They include:
- **Change of Base Rule**: Helps converting logarithms of different bases, useful if calculators have specific log functions.
- **Single Logarithm Rule**: Helps compress multiple logarithms into one, which simplifies equations.
Consider \(\text{log}_2(x) - \text{log}_2(3) = 5\). Using the quotient rule, rewrite it to get \(\text{log}_2\left( \frac{x}{3} \right) = 5\). Then, convert to exponential form \(\frac{x}{3} = 2^5\), which becomes \(\frac{x}{3} = 32\). Solve for \(x\) by multiplying both sides by 3, resulting in \(x = 96\).

Rewriting logarithms in exponential form is crucial. It helps in solving complex logarithmic equations. The general transform is \(\text{log}_b(a) = c\) converts to \(b^c = a\).
For example, with \(\text{log}_6(x) = 2 - \text{log}_6(4)\), use subtraction rule to combine terms: \(\text{log}_6(x) = \text{log}_6 \left( \frac{36}{4} \right)\), simplifying to \(\text{log}_6(x) = \text{log}_6(9)\). Since the logs are equal, \(x = 9\). The conversion to exponential form helps to isolate and solve for the variable directly when dealing with more complex logarithmic components.