Math Snap

STEP 1

Assumptions1. The base of all logarithms in the equation is.
. The variable x is the same in all parts of the equation.
3. The logarithmic equation is valid for x >0 and x ≠1.

STEP 2

We can use the properties of logarithms to simplify the equation. The property $\log_b a - \log_b c = \log_b \frac{a}{c}$ allows us to combine the left side of the equation.
$$\log{2} x-\log{2}7=\log{2}(x-1)$$becomes$$\log{2} \frac{x}{7} = \log{2}(x-1)$$

STEP 3

Since the base of the logarithms on both sides of the equation is the same (base2), we can set the arguments of the logarithms equal to each other.
$$\frac{x}{7} = x -1$$

STEP 4

To solve for x, we first get rid of the fraction by multiplying every term by7.
$$x =7x -7$$

STEP 5

Next, we rearrange the equation to isolate x. Subtract x from both sides to get$$0 =x -7$$

STEP 6

Add to both sides to isolate the term with x$$=6x$$

Finally, divide both sides by6 to solve for x$$x = \frac{7}{6}$$So, the solution to the logarithmic equation $\log{2} x-\log{2}7=\log{2}(x-1)$ is $x = \frac{7}{6}$.