Chapter 5: Q.76 (page 316)

solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places:
$\log_{2} (3 x + 2) - \log_{4} (x) = 3$

Short Answer

Expert verified

The value of x satisfying the equation is $x = 0.078 a n d x = 5.69$

Step by step solution

Given information

We are given a equation $\log_{2} (3 x + 2) - \log_{4} (x) = 3$

Change the base

We have,

$\log_{2} (3 x + 2) - \log_{4} (x) = 3 \log_{2} (3 x + 2) - \frac{\log_{2} (x)}{\log_{2} {(2)}^{2}} = 3 \log_{2} (3 x + 2) - \frac{\log_{2} (x)}{2} = 3 2 \log_{2} (3 x + 2) - \log_{2} (x) = 6$

Solve the equation

We get,

$2 \log_{2} (3 x + 2) - \log_{2} (x) = 6 l {og}_{2} {(3 x + 2)}^{2} - \log_{2} (x) = 6 \log_{2} \frac{{(3 x + 2)}^{2}}{x} = 6$

Now use the formula $y = \log_{a} (x) \Leftrightarrow x = a^{y}$

$\frac{{(3 x + 2)}^{2}}{x} = 2^{6} {(3 x + 2)}^{2} = 64 x 9 x^{2} + 12 x + 4 = 64 x 9 x^{2} - 52 x + 4 = 0$

Find the value of x

We get,

$9 x^{2} - 52 x + 4 = 0 x = \frac{52 \pm \sqrt{52^{2} - 4 (9) (4)}}{18} x = \frac{52 \pm \sqrt{2704 - 144}}{18} x = \frac{52 \pm 50.59}{18} x = \frac{52 + 50.59}{18} o r x = \frac{52 - 50.59}{18} x = 5.69 o r x = 0.078$

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solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places:
$\log_{2} (3 x + 2) - \log_{4} (x) = 3$

Short Answer

Step by step solution

Given information

Change the base

Solve the equation

Find the value of x

One App. One Place for Learning.

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solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places: log2(3x+2)-log4(x)=3

Short Answer

Step by step solution

Given information

Change the base

Solve the equation

Find the value of x

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Applied Mathematics

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places:
$\log_{2} (3 x + 2) - \log_{4} (x) = 3$