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: \(\log_b A + \log_b B = \log_b (A \times B)\). Apply this to combine the left side: \(\log_3 (x+6) + \log_3 (x+4) = \log_3 \big((x+6)(x+4)\big)\).
2. Rewrite the equation using the combined logarithm: \(\log_3 \big((x+6)(x+4)\big) = 1\).
3. Convert the logarithmic equation to its equivalent exponential form. Recall that \(\log_b y = c\) means \(b^c = y\). So, rewrite as: \$3^1 = (x+6)(x+4)$.
4. Simplify the right side by expanding the product: \((x+6)(x+4) = x^2 + 4x + 6x + 24 = x^2 + 10x + 24\). So the equation becomes \$3 = x^2 + 10x + 24$.
5. Bring all terms to one side to set the quadratic equation to zero: \(x^2 + 10x + 24 - 3 = 0\), which simplifies to \(x^2 + 10x + 21 = 0\). Then solve this quadratic equation using factoring, completing the square, or the quadratic formula.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, especially the product rule, is essential. The product rule states that log_b(A) + log_b(B) = log_b(AB), allowing the combination of two logarithmic expressions into one. This simplifies solving equations by converting sums of logs into a single log expression.

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 the solutions do not make any log argument zero or negative, as these values are not defined in the real number system.

Solving Exponential Equations

After applying logarithmic properties, the equation often converts to an exponential form. Solving the resulting exponential equation involves isolating the variable and finding exact values. This step is key to determining the solution before verifying domain restrictions.