〖log〗_5⁡(4x+1)-〖log〗_5⁡(2x+10)=0

To solve the equation:

\[
\log_5(4x + 1) - \log_5(2x + 10) = 0
\]

### Step 1: Use the logarithmic property
The property of logarithms we can apply here is:

\[
\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)
\]

Using this on the given equation:

\[
\log_5 \left(\frac{4x + 1}{2x + 10}\right) = 0
\]

### Step 2: Convert to exponential form
Recall that \(\log_b C = 0\) means \(C = 1\). Thus, we can rewrite the equation as:

\[
\frac{4x + 1}{2x + 10} = 1
\]

### Step 3: Solve the equation
Multiply both sides by \(2x + 10\) to eliminate the denominator:

\[
4x + 1 = 2x + 10
\]

Now, subtract \(2x\) from both sides:

\[
2x + 1 = 10
\]

Next, subtract 1 from both sides:

\[
2x = 9
\]

Finally, divide by 2:

\[
x = \frac{9}{2}
\]

### Step 4: Check for extraneous solutions
Substitute \(x = \frac{9}{2}\) back into the original logarithmic expressions to check for validity:

\[
4x + 1 = 4 \times \frac{9}{2} + 1 = 18 + 1 = 19
\]
\[
2x + 10 = 2 \times \frac{9}{2} + 10 = 9 + 10 = 19
\]

Since both expressions are positive, \(x = \frac{9}{2}\) is valid.

### Final Solution:
\[
x = \frac{9}{2}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you solve logarithmic equations involving different bases?
2. What happens if the argument inside a logarithm becomes negative?
3. Can logarithms with subtraction be combined in other ways?
4. How do you graph logarithmic functions?
5. How does the change-of-base formula work in logarithms?

**Tip:** Always check the domain of logarithmic functions to ensure the argument is positive!

Learn how to solve the logarithmic equation log_5(4x + 1) - log_5(2x + 10) = 0 step by step. This solution covers key concepts in logarithms, such as logarithmic properties and converting logarithmic to exponential form, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation