Chapter 5: Q.75 (page 316)

solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places:
$\log_{2} (x + 1) - \log_{4} x = 1$

Short Answer

Expert verified

Solution of the equation is $x = 1$

Step by step solution

Given information

We are given an equation

$\log_{2} (x + 1) - \log_{4} (x) = 1$

Change the base

We get,

$\log_{2} (x + 1) - \log_{4} (x) = 1 \log_{2} (x + 1) - \frac{\log_{2} (x)}{\log_{2} (4)} = 1 \log_{2} (x + 1) - \frac{\log_{2} (x)}{\log_{2} {(2)}^{2}} = 1 \log_{2} (x + 1) - \frac{\log_{2} (x)}{2} = 1 2 \log_{2} (x + 1) - \log_{2} (x) = 2$

Solve the equation

We get,

$2 \log_{2} (x + 1) - \log_{2} (x) = 2 \log_{2} {(x + 1)}^{2} - \log_{2} (x) = 2 \log_{2} (\frac{{(x + 1)}^{2}}{x}) = 2$

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

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

Conclusion

Solution of the above equation is $x = 1$

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solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places:
$\log_{2} (x + 1) - \log_{4} x = 1$

Short Answer

Step by step solution

Given information

Change the base

Solve the equation

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

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solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places:log2x+1-log4x=1

Short Answer

Step by step solution

Given information

Change the base

Solve the equation

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Calculus

Pure Maths

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places:
$\log_{2} (x + 1) - \log_{4} x = 1$