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 log₃(x+4)=log₃ 9+2

Verified step by step guidance

1. Start by rewriting the given equation: $2\log 3_{}(x+4)=\log 3_{}9+2$.
2. Use the logarithm power rule on the left side: $2\log 3_{}(x+4)=\log 3_{}(x+4)2^{}$, so rewrite the equation as $\log 3_{}(x+4{)}^2=\log 3_{}9+2$.
3. Rewrite the constant term 2 on the right side as a logarithm with base 3: since $2\; =\; log\_3(3^2)\; =\; log\_3(9)$, the right side becomes $log\_3(9)\; +\; log\_3(9)$.
4. Use the logarithm addition rule on the right side: $log\_3(9)\; +\; log\_3(9)\; =\; log\_3(9\; imes\; 9)\; =\; log\_3(81)$.
5. Now the equation is $log\_3((x+4)^2)\; =\; log\_3(81)$. Since the logs are equal and the base is the same, set the arguments equal: $(x+4)^2\; =\; 81$. Then solve this quadratic equation for $x$, remembering to check the domain restrictions for the original logarithmic expressions (i.e., $x+4\; >\; 0$).

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 power rule allows you to move coefficients in front of the log as exponents, which helps in rewriting and solving the equation.

Domain of Logarithmic Functions

The domain of a logarithmic function includes all values for which the argument (inside the log) is positive. When solving logarithmic equations, it is crucial to check that solutions do not make any log argument zero or negative, as these are not valid in the real number system.

Converting Logarithmic Equations to Exponential Form

To solve logarithmic equations, it is often helpful to rewrite them in exponential form. This conversion allows you to solve for the variable more directly by removing the logarithm, making it easier to isolate and find the exact value of the unknown.