Question

which equation(s) have \\(x = 2\\) as the solution?

\\(\log_{2}(5x + 6) = 4\\)
\\(\log_{x}16 = 4\\)
\\(\log_{3}(6x + 4) = 3\\)
\\(\log_{x}36 = 6\\)

Explanation:

⚡ Using what you learned: Logarithms and Exponential Equations

Step 1: Test the first equation

Substitute \( x = 2 \) into \( \log_{2}(5x + 6) = 4 \):

$$ \log_{2}(5(2) + 6) = \log_{2}(10 + 6) = \log_{2}(16) $$

Since \( 2^4 = 16 \):

$$ \log_{2}(16) = 4 $$

This equation is true for \( x = 2 \).

Step 2: Test the second equation

Substitute \( x = 2 \) into \( \log_{x}16 = 4 \):

$$ \log_{2}16 = 4 $$

Since \( 2^4 = 16 \):

$$ 4 = 4 $$

This equation is true for \( x = 2 \).

Step 3: Test the third equation

Substitute \( x = 2 \) into \( \log_{3}(6x + 4) = 3 \):

$$ \log_{3}(6(2) + 4) = \log_{3}(12 + 4) = \log_{3}(16) $$

Since \( 3^3 = 27
eq 16 \):

$$ \log_{3}(16) eq 3 $$

This equation is false for \( x = 2 \).

Step 4: Test the fourth equation

Substitute \( x = 2 \) into \( \log_{x}36 = 6 \):

$$ \log_{2}36 = 6 $$

Since \( 2^6 = 64
eq 36 \):

$$ \log_{2}36 eq 6 $$

This equation is false for \( x = 2 \).

Answer:

The equations that have \( x = 2 \) as a solution are:

• \( \log_{2}(5x + 6) = 4 \)
• \( \log_{x}16 = 4 \)