Chapter 10: Q. 10.80 (page 1049)

Solve: $\log (x - 2) - \log (4 x + 16) = \log \frac{1}{x}$

Short Answer

Expert verified

By solving the equation, $\log (x - 2) - \log (4 x + 16) = \log \frac{1}{x}$ , the value of $x$ is $8$

Step by step solution

Step 1. Given

An expression $\log (x - 2) - \log (4 x + 16) = \log \frac{1}{x}$

To solve the expression

Step 2. Use the Quotient Property on the left side

Use the Quotient Property on the left side

$\log (x - 2) - \log (4 x + 16) = \log \frac{1}{x} \log (\frac{x - 2}{4 x + 16}) = \log \frac{1}{x}$

Step 3. Use the One-to-One Property,

Step 3. Use the One-to-One Property, if $\log_{a} M = \log_{a} N$ , then $M = N$
$(\frac{x - 2}{4 x + 16}) = \frac{1}{x} x^{2} - 2 x = 4 x + 16 x^{2} - 6 x - 16 = 0$

Step 4. Factorize the equation

$x^{2} - 6 x - 16 = 0 (x - 8) (x + 2) = 0 x = - 2, 8$

Logarithm for a negative number does not exist.

So, $x = 8$

Step 5. Check the solution

Substitute $x = 8$ in the original equation and solve.

$\log (8 - 2) - \log (4 (8) + 16) = \log \frac{1}{8} \log (6) - \log (48) = \log \frac{1}{8} \log (\frac{6}{48}) = \log \frac{1}{8} \frac{1}{8} = \frac{1}{8}$

Hence the solution has been checked.

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Solve: $\log (x - 2) - \log (4 x + 16) = \log \frac{1}{x}$

Short Answer

Step by step solution

Step 1. Given

Step 2. Use the Quotient Property on the left side

Step 3. Use the One-to-One Property,

Step 4. Factorize the equation

Step 5. Check the solution

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Solve: log(x-2)-log(4x+16)=log1x

Short Answer

Step by step solution

Step 1. Given

Step 2. Use the Quotient Property on the left side

Step 3. Use the One-to-One Property,

Step 4. Factorize the equation

Step 5. Check the solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Solve: $\log (x - 2) - \log (4 x + 16) = \log \frac{1}{x}$