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)$

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Identify the properties of logarithms that will help condense the expression: the product rule $\log a + \log b = \log(ab)$ and the quotient rule $\log a - \log b = \log\left(\frac{a}{b}\right)$.

Apply the product rule to combine the positive logarithms: $\log x + \log(x^2 - 1) = \log\left(x(x^2 - 1)\right)$.

Apply the quotient rule to combine the negative logarithms: $- \log 7 - \log(x + 1) = - \log(7(x + 1)) = \log\left(\frac{1}{7(x + 1)}\right)$.

Combine the results from steps 2 and 3 using the quotient rule: $\log\left(x(x^2 - 1)\right) + \log\left(\frac{1}{7(x + 1)}\right) = \log\left(\frac{x(x^2 - 1)}{7(x + 1)}\right)$.

Recognize that $x^2 - 1$ is a difference of squares and factor it as $(x - 1)(x + 1)$, then simplify the expression inside the logarithm by canceling common factors.

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Key Concepts

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

Properties of Logarithms

Properties of logarithms include rules such as the product rule (log a + log b = log(ab)), the quotient rule (log a - log b = log(a/b)), and the power rule (k log a = log(a^k)). These allow combining or breaking down logarithmic expressions to simplify or condense them.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves factoring and reducing expressions inside the logarithms. For example, recognizing that x^2 - 1 factors as (x - 1)(x + 1) helps in canceling terms when combined with other logarithms, making the expression easier to condense.

Evaluating Logarithmic Expressions Without a Calculator

Evaluating logarithmic expressions without a calculator requires recognizing values that simplify to known logarithms, such as log 1 = 0 or log of perfect powers. This skill helps in simplifying the final expression or determining if it can be further reduced to a numerical value.

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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

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)

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