Math Problem Statement

Solve the following logarithmic equation. Express irrational solutions in exact form. log (8x + 1) = 1 + log (x - 4) Rewrite the given equation without logarithms. Do not solve for x.

Solution

The given logarithmic equation is:

log(8x+1)=1+log(x4)\log(8x + 1) = 1 + \log(x - 4)

Step 1: Rewrite the equation without logarithms.

We can start by isolating the logarithmic terms on one side:

log(8x+1)log(x4)=1\log(8x + 1) - \log(x - 4) = 1

Using the logarithmic difference rule, log(a)log(b)=log(ab)\log(a) - \log(b) = \log\left(\frac{a}{b}\right), we get:

log(8x+1x4)=1\log\left(\frac{8x + 1}{x - 4}\right) = 1

Step 2: Eliminate the logarithm.

To eliminate the logarithm, we raise both sides of the equation as powers of 10 (since log\log is base 10):

8x+1x4=101=10\frac{8x + 1}{x - 4} = 10^1 = 10

This is the rewritten form of the equation without logarithms.

Would you like me to continue solving for xx, or do you have any other questions about the steps so far?


Here are 5 related questions for deeper understanding:

  1. What are the conditions for a logarithmic function to be defined?
  2. How do you solve rational equations like 8x+1x4=10\frac{8x + 1}{x - 4} = 10?
  3. How does the logarithmic difference rule work in other cases?
  4. How do we express irrational solutions in their exact form?
  5. Can you explain the concept of logarithmic bases and their importance in solving equations?

Tip: When solving logarithmic equations, always check for extraneous solutions by ensuring the arguments of the logarithms remain positive.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Equations
Algebra

Formulas

log(a) - log(b) = log(a/b)
a = 10^b for base-10 logarithms

Theorems

Properties of Logarithms

Suitable Grade Level

Grades 9-12