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. log₃(x+6)+log₃(x+4)=1

Verified step by step guidance

1. Recall the logarithmic property that allows you to combine the sum of two logarithms with the same base: $\log 3_{}\left(a\right)+\log 3_{}\left(b\right)=\log 3_{}(a\cdot b)$. Use this to combine the left side of the equation.
2. Apply the property to rewrite the equation as $\log 3_{}\left(x+6\right)+\log 3_{}\left(x+4\right)=\log 3_{}((x+6)\; (x+4)$, so the equation becomes $\log 3_{}((x+6)\; (x+4)$ = 1.
3. Rewrite the right side of the equation in logarithmic form: since $1\; =\; log\_3(3)$, the equation becomes $\log 3_{}((x+6)\; (x+4)$ = $\log 3_{}\left(3\right)$.
4. Since the logarithms with the same base are equal, set their arguments equal: $(x+6)\; (x+4)\; =3$. Expand the left side to form a quadratic equation.
5. Solve the quadratic equation for $x$. After finding the solutions, check each one to ensure it does not make the arguments of the original logarithms negative or zero, because the domain of $log\_3(x)$ requires $x\; >\; 0$.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product rule log_b(m) + log_b(n) = log_b(mn), is essential for combining or simplifying logarithmic expressions. This allows the equation to be rewritten in a simpler form, making it easier to solve.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) requires that the argument x be 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 not valid in the real number system.

Solving Exponential Equations

After applying logarithmic properties, the equation often converts to an exponential form. Solving exponential equations involves isolating the variable and using inverse operations, such as raising both sides to the base of the logarithm, to find exact solutions.