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. log₂(x+2)−log₂(x−5)=3

Verified step by step guidance

1. Recall the logarithmic property that states: $\log {\mathrm{a}}_{\mathrm{b}}-\log {\mathrm{a}}_c=\log {\mathrm{a}}_{\frac{b}{c}}$. Apply this to combine the left side: $log\_2(x+2)\; -\; log\_2(x-5)\; =\; log\_2\backslash left(\backslash frac\{x+2\}\{x-5\}\backslash right)$.
2. Rewrite the equation using the property: $log\_2\backslash left(\backslash frac\{x+2\}\{x-5\}\backslash right)\; =\; 3$. This means that the argument of the logarithm equals the base raised to the power on the right side: $\backslash frac\{x+2\}\{x-5\}\; =\; 2^3$.
3. Calculate the right side exponent: $2^3\; =\; 8$. So, the equation becomes $\backslash frac\{x+2\}\{x-5\}\; =\; 8$.
4. Solve the equation for $x$ by cross-multiplying: $x\; +\; 2\; =\; 8(x\; -\; 5)$. Then expand and simplify: $x\; +\; 2\; =\; 8x\; -\; 40$.
5. Isolate $x$ by bringing all terms to one side: $2\; +\; 40\; =\; 8x\; -\; x$, which simplifies to $42\; =\; 7x$. Then solve for $x$: $x\; =\; \backslash frac\{42\}\{7\}$. Finally, check the domain restrictions to ensure the solution is valid: the arguments of the original logarithms must be positive, so $x\; +\; 2\; >\; 0$ and $x\; -\; 5\; >\; 0$.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the difference rule log_b(A) - log_b(B) = log_b(A/B), is essential for simplifying and solving logarithmic equations. These properties allow combining or breaking down logarithmic expressions to isolate the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function includes all values for which the argument is positive. When solving logarithmic equations, it is crucial to check that solutions do not make any logarithm’s argument zero or negative, as these are undefined.

Solving Exponential Equations

After applying logarithmic properties, the equation often converts to an exponential form. Solving this exponential equation involves isolating the variable and finding exact values, which may then be approximated using a calculator if needed.