Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2 log x=log 25

Verified step by step guidance

1. Start with the given equation: $2\log x=\log 25$. Recognize that the logarithm without a base typically means base 10 (common logarithm).
2. Use the logarithmic property that allows you to move the coefficient in front of the log as an exponent inside the log: $2\; \backslash log\; x\; =\; \backslash log\; x^\{2\}$. So rewrite the equation as $\backslash log\; x^\{2\}\; =\; \backslash log\; 25$.
3. Since the logs on both sides have the same base and are equal, set the arguments equal to each other: $x^\{2\}\; =\; 25$.
4. Solve the equation $x^\{2\}\; =\; 25$ by taking the square root of both sides, remembering to consider both positive and negative roots: $x\; =\; \backslash pm\; 5$.
5. Check the domain of the original logarithmic expression: since $\backslash log\; x$ is defined only for $x\; >\; 0$, reject any solution that does not satisfy this. Therefore, exclude $x\; =\; -5$ and keep $x\; =\; 5$ as the valid solution.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential for simplifying and solving logarithmic equations. For example, the equation 2 log x = log 25 can be rewritten using the power rule as log(x^2) = log 25, which helps in isolating the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive real numbers because the logarithm of zero or a negative number is undefined. When solving logarithmic equations, it is crucial to check that the solutions fall within the domain to reject any extraneous or invalid answers.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation in exponential form to isolate the variable. For instance, from log(x^2) = log 25, we deduce x^2 = 25, then solve for x, considering domain restrictions. This approach leads to exact solutions before approximating decimals.