Chapter 10: Q. 10.79 (page 1049)

Solve: $\log (x + 2) - \log (4 x + 3) = - \log x$

Short Answer

Expert verified

By solving the equation, $\log (x + 2) - \log (4 x + 3) = - \log x$ , the value of $x$ is $3$

Step by step solution

Step 1. Given

An expression $\log (x + 2) - \log (4 x + 3) = - \log x$

To solve the expression

Step 2. Use the Quotient Property on the left side and the Power Property on the right.

Use the Quotient Property on the left side and the Power Property on the right.

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

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

Use the One-to-One Property, if $\log_{a} M = \log_{a} N$ , then $M = N$

$\frac{x + 2}{4 x + 3} = \frac{1}{x} x^{2} + 2 x = 4 x + 3 x^{2} - 2 x - 3 = 0$

Step 4. Factorize the equation

$x^{2} - 2 x - 3 = 0 (x - 3) (x + 1) = 0 x = - 1, 3$

Logarithm for a negative number does not exist.

So, $x = 3$

Step 5. Check the solution

Substitute $x = 3$ in the original equation and solve.

$\log (3 + 2) - \log (4 (3) + 3) = - \log 3 \log (5) - \log (15) = \log \frac{1}{3} \log (\frac{5}{15}) = \log \frac{1}{3} \frac{1}{3} = \frac{1}{3}$

Hence the solution has been checked.

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Solve: $\log (x + 2) - \log (4 x + 3) = - \log x$

Short Answer

Step by step solution

Step 1. Given

Step 2. Use the Quotient Property on the left side and the Power Property on the right.

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+3)=-logx

Short Answer

Step by step solution

Step 1. Given

Step 2. Use the Quotient Property on the left side and the Power Property on the right.

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

Geometry

Calculus

Logic and Functions

Discrete Mathematics

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Solve: $\log (x + 2) - \log (4 x + 3) = - \log x$