Math Problem Statement

Solve log_4(x - 2) = 1.

Solution

To solve the given equation:

log4(x2)=1\log_4(x - 2) = 1

We proceed as follows:

Step 1: Convert logarithmic to exponential form

The general formula for converting logarithmic equations is:

logb(a)=c    bc=a\log_b(a) = c \quad \implies \quad b^c = a

Here, b=4b = 4, c=1c = 1, and a=x2a = x - 2. Applying the formula:

41=x24^1 = x - 2

Step 2: Simplify the equation

4=x24 = x - 2

Step 3: Solve for xx

Add 2 to both sides: x=4+2x = 4 + 2

x=6x = 6

Final Answer:

The solution is x=6x = 6. Therefore, the correct option is:

D. {6}


Would you like a detailed explanation of logarithms, or do you have further questions? Here are some related questions for practice:

  1. Solve log3(x+1)=2\log_3(x + 1) = 2.
  2. If log2(x3)=4\log_2(x - 3) = 4, find xx.
  3. Rewrite the equation log7(49)=2\log_7(49) = 2 in exponential form.
  4. Explain the domain of logb(x)\log_b(x).
  5. Solve log5(x)=1\log_5(x) = -1.

Tip: Always ensure that the argument of the logarithmic function (e.g., x2x - 2) is positive, as logarithms are undefined for non-positive values.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponential Equations

Formulas

log_b(a) = c ⟺ b^c = a

Theorems

Logarithmic to exponential conversion

Suitable Grade Level

Grades 9-11