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

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: $\log_3 (x+6) + \log_3 (x+4) = \log_3 \big((x+6)(x+4)\big)$.

Rewrite the equation using the combined logarithm: $\log_3 \big((x+6)(x+4)\big) = 1$.

Convert the logarithmic equation to its equivalent exponential form. Recall that $\log_b y = c$ means $b^c = y$. So, rewrite as: $3^1 = (x+6)(x+4)$.

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 $3 = x^2 + 10x + 24$.

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Recommended video:

5:47

Solving Exponential Equations Using Logs

Related Practice

Textbook Question

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

Textbook Question

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log₅13

Textbook Question

In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $8^{x} = 12143$

Textbook Question

The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. h(x) = ln(x/2)

Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. $$\log$ x + $\log$(x^2 - 1) - $\log$ 7 - $\log$(x + 1)$

Textbook Question