Question

which equation is equivalent to \\(\log_{3}(x+5) = 2\\)?

• \\(3^{2} = \log_{3}(x+5)^{3}\\)
• \\(2^{3} = \log_{3}(x+5)^{2}\\)
• \\(3^{2} = x+5\\)
• \\(2^{3} = x+5\\)

Explanation:

Identify the logarithmic equation

The given equation is:

$$ \log_{3}(x+5) = 2 $$

Apply the definition of a logarithm

Using the definition of a logarithm, \(\log_{b}(a) = c\) is equivalent to the exponential form:

$$ b^c = a $$

Here, the base is \(b = 3\), the argument is \(a = x+5\), and the exponent is \(c = 2\).

Convert to exponential form

Substituting these values into the exponential form yields:

$$ 3^2 = x+5 $$

Answer:

• (A) \(3^2 = [\log_3(x+5)]^3\)
• (B) \(2^3 = [\log_3(x+5)]^2\)
• (C) \(3^2 = x+5\) (Correct answer)
• (D) \(2^3 = x+5\)