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Question
Solve the equation for x.

log2(x−3)−log23 = 5

(1 point)
x =

1 answer

To solve the equation \( \log_2(x - 3) - \log_2(3) = 5 \), we can use the properties of logarithms. Specifically, we will use the property that \( \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right) \).

Using this property, we can rewrite the equation as:

\[ \log_2\left(\frac{x - 3}{3}\right) = 5 \]

Now, to eliminate the logarithm, we will exponentiate both sides, using the fact that if \( \log_a(b) = c \), then \( b = a^c \):

\[ \frac{x - 3}{3} = 2^5 \]

Calculating \( 2^5 \):

\[ 2^5 = 32 \]

Thus, we have:

\[ \frac{x - 3}{3} = 32 \]

Now we can solve for \( x \) by multiplying both sides by 3:

\[ x - 3 = 96 \]

Next, we add 3 to both sides:

\[ x = 96 + 3 = 99 \]

So the solution is:

\[ \boxed{99} \]