What is the value of x in the equation log_2(x - 5) + log_7(1) = log_3(27)?

To find the value of x in the equation log_2(x - 5) + log_7(1) = log_3(27), we start by simplifying the components of the equation.

Firstly, we recognize that log_7(1) = 0 because any logarithm of 1 to any base is 0. Thus, the equation simplifies to:

log_2(x - 5) + 0 = log_3(27)

This reduces to:

log_2(x - 5) = log_3(27)

Next, we need to find the value of log_3(27). We can rewrite 27 as 3^3. Therefore:

log_3(27) = log_3(3^3) = 3

Now our equation reads:

log_2(x - 5) = 3

To remove the logarithm, we can rewrite this in exponential form. The equation log_2(x - 5) = 3 means that:

x - 5 = 2^3

Calculating 2^3, we get:

x - 5 = 8

To find x, we will add 5 to both sides of the equation