Q. 434 In the following exercises, solv... [FREE SOLUTION]

91影视

Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 10: Q. 434 (page 1065)

In the following exercises, solve for x.
$\log (x - 1) - \log (3 x + 5) = - \log x$

Short Answer

Expert verified

The value of $x$ for the equation role="math" localid="1645038702854" $\log (x - 1) - \log (3 x + 5) = - \log x$ is $5$

Step by step solution

Step 1. Given data

The given Logarithmic Equation is $\log (x - 1) - \log (3 x + 5) = - \log x$

Step 2. Apply the quotient properties of the logarithm

Use quotient properties of the logarithm on the left side and power property on the right side

$\log (x - 1) - \log (3 x + 5) = - \log x \log \frac{(x - 1)}{(3 x + 5)} = \log x^{- 1} \log \frac{(x - 1)}{(3 x + 5)} = \log \frac{1}{x}$

Step 3. Apply one to one property

Use one to one property of logarithms, if $\log_{a} (m) = \log_{a} (n)$

then $m = n$

so $\log \frac{(x - 1)}{(3 x + 5)} = \log \frac{1}{x} \frac{(x - 1)}{(3 x + 5)} = \frac{1}{x} x (x - 1) = (3 x + 5) x^{2} - x = 3 x + 5 x^{2} - 4 x - 5 = 0 (x + 1) (x - 5) = 0$

so $x = 5, - 1$

the logarithm of a negative number is not possible

so $x = 5$

Step 4. verification

Substitute $5$ for x in the equation

$\log (x - 1) - \log (3 x + 5) = - \log x \log (5 - 1) - \log (3 (5) + 5) \overset{?}{=} - \log 5 \log 4 - \log (15 + 5) \overset{?}{=} \log 5^{- 1} \log 4 - \log 20 \overset{?}{=} \log \frac{1}{5} \log (\frac{4}{20}) \overset{?}{=} \log \frac{1}{5} \log \frac{1}{5} = \log \frac{1}{5}$

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91影视

In the following exercises, solve for x.
$\log (x - 1) - \log (3 x + 5) = - \log x$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Apply the quotient properties of the logarithm

Step 3. Apply one to one property

Step 4. verification

One App. One Place for Learning.

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91影视

In the following exercises, solve for x.logx-1-log3x+5=-logx

Short Answer

Step by step solution

Step 1. Given data

Step 2. Apply the quotient properties of the logarithm

Step 3. Apply one to one property

Step 4. verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

In the following exercises, solve for x.
$\log (x - 1) - \log (3 x + 5) = - \log x$