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Which equation is equivalent to [tex]\log _3(x+5)=2[/tex]?

A. [tex]3^2=\left[\log _3(x+5)\right]^3[/tex]

B. [tex]2^3=\left[\log _3(x+5)\right]^2[/tex]

C. [tex]3^2=x+5[/tex]

D. [tex]2^3=x+5[/tex]

Sagot :

To solve the given problem, let's start by analyzing the logarithmic equation [tex]\(\log_3(x + 5) = 2\)[/tex]. Our goal is to find an equivalent equation in a different form.


1. Understanding the Logarithmic Equation:
The given equation is [tex]\(\log_3(x + 5) = 2\)[/tex]. A logarithm is the inverse of an exponentiation. Specifically, [tex]\(\log_b(a) = c\)[/tex] means that the base [tex]\(b\)[/tex] raised to the power of [tex]\(c\)[/tex] equals [tex]\(a\)[/tex].

2. Convert to Exponential Form:
The equation [tex]\(\log_3(x + 5) = 2\)[/tex] can be rewritten in its exponential form as follows:
[tex]\[ 3^2 = x + 5 \][/tex]
This step uses the property that if [tex]\(\log_b(a) = c\)[/tex], then [tex]\(b^c = a\)[/tex].

3. Rewrite the Equation:
Simplify [tex]\(3^2\)[/tex]. We know that:
[tex]\[ 3^2 = 9 \][/tex]
Hence, the equation [tex]\(3^2 = x + 5\)[/tex] simplifies to:
[tex]\[ 9 = x + 5 \][/tex]

4. Shifting Terms:
To isolate [tex]\(x\)[/tex], we subtract 5 from both sides of the equation:
[tex]\[ 9 - 5 = x \][/tex]
Simplifying further gives:
[tex]\[ 4 = x \][/tex]

Among the provided options, the equivalent equation to [tex]\(\log_3(x + 5) = 2\)[/tex] is:
[tex]\[ 3^2 = x + 5 \][/tex]

Therefore, the correct answer is:
[tex]\[ 3^2 = x + 5 \][/tex]