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. 4 ln (x + 6) - 3 ln x

Verified step by step guidance

Identify the logarithmic expression given: \(4 \ln (x + 6) - 3 \ln x\).

Use the power rule of logarithms, which states that \(a \ln b = \ln (b^a)\), to rewrite each term: \(4 \ln (x + 6) = \ln ((x + 6)^4)\) and \(3 \ln x = \ln (x^3)\).

Rewrite the expression using these powers: \(\ln ((x + 6)^4) - \ln (x^3)\).

Apply the logarithmic subtraction rule, which states \(\ln A - \ln B = \ln \left( \frac{A}{B} \right)\), to combine the terms into a single logarithm: \(\ln \left( \frac{(x + 6)^4}{x^3} \right)\).

The expression is now condensed into a single logarithm with coefficient 1: \(\ln \left( \frac{(x + 6)^4}{x^3} \right)\). No further simplification is possible without specific values for \(x\).

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

Properties of logarithms include rules such as the product, quotient, and power rules. These allow combining or breaking down logarithmic expressions. For example, the power rule lets you move coefficients as exponents inside the log, which is essential for condensing expressions.

Power Rule of Logarithms

The power rule states that a coefficient in front of a logarithm can be rewritten as an exponent inside the log: a * ln(b) = ln(b^a). This is crucial for rewriting expressions like 4 ln(x + 6) as ln((x + 6)^4) to combine terms into a single logarithm.

Combining Logarithmic Expressions

Using the product and quotient rules, multiple logarithms can be combined into one. The product rule states ln(a) + ln(b) = ln(ab), and the quotient rule states ln(a) - ln(b) = ln(a/b). Applying these helps write the expression as a single logarithm with coefficient 1.

Recommended video:

7:30

Logarithms Introduction

Related Practice

Textbook Question

In Exercises 60–63, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. (ln x)(ln 1) = 0

Textbook Question

Give the equation of each exponential function whose graph is shown.

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. log₄(3x+2)=3

Textbook Question

In Exercises 58–59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log₄ 0.863

Textbook Question

The figure shows the graph of f(x) = log x. In Exercises 59–64, 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. g(x) = log(x − 1)