Chapter 5: Q 55. (page 308)

Write each expression as a sum and/or difference of logarithms. Express powers as factors.
$\ln \frac{5 x \sqrt{1 + 3 x}}{{(x - 4)}^{3}}$

Short Answer

Expert verified

The result is $\ln 5 + \ln x + \frac{1}{2} \ln (1 + 3 x) + 3 \ln (x - 4)$ .

Step by step solution

Step 1. Given information.

Consider the given information.

$\ln \frac{5 x \sqrt{1 + 3 x}}{{(x - 4)}^{3}}$

Step 2. Formula used.

The logarithm properties are defined as,

$\begin{array}{l} \log_{a} (\frac{M}{N}) = \log_{a} M - \log_{a} N \\ \log_{a} (M N) = \log_{a} M + \log_{a} N \\ \log_{a} M^{r} = r \log_{a} M \end{array}$

Step 3. Calculation.

Apply the logarithm property to simplify.

$\begin{array}{rcl} \ln \frac{5 x \sqrt{1 + 3 x}}{{(x - 4)}^{3}} & = & \ln (5 x \sqrt{1 + 3 x}) + \ln {(x - 4)}^{3} \\ = & \ln 5 x + \ln (\sqrt{1 + 3 x}) + 3 \ln (x - 4) \\ = & \ln 5 + \ln x + \ln {(1 + 3 x)}^{\frac{1}{2}} + 3 \ln (x - 4) \\ = & \ln 5 + \ln x + \frac{1}{2} \ln (1 + 3 x) + 3 \ln (x - 4) \end{array}$

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Write each expression as a sum and/or difference of logarithms. Express powers as factors.
$\ln \frac{5 x \sqrt{1 + 3 x}}{{(x - 4)}^{3}}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Calculation.

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Write each expression as a sum and/or difference of logarithms. Express powers as factors.ln5x1+3xx-43

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Calculation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Write each expression as a sum and/or difference of logarithms. Express powers as factors.
$\ln \frac{5 x \sqrt{1 + 3 x}}{{(x - 4)}^{3}}$