Solve the logarithmic equation for x. (Enter your answers as a comma-separated list.)
log2(x + 17) − log2(x − 2) = 1
2 is the base for the log.
using our rules of logs , which you should have at your fingertips,
log2 ( (x+17)/(x-2) ) = 1
so
(x+17)/(x-2) = 2^1
solve for x , remember x> 2 for the log to be defined.
Oh logarithmic equations, always trying to be sneaky. Alright, let's solve this one step at a time.
First, let's use a little log magic to simplify the equation. We can use the quotient rule for logarithms, which says that log(A) - log(B) is equal to log(A/B). So, our equation becomes:
log2((x + 17)/(x - 2)) = 1
Now, we can use the fact that log2(2^k) = k to rewrite the equation as an exponential equation:
2^1 = (x + 17)/(x - 2)
Simplifying further:
2 = (x + 17)/(x - 2)
Now, let's get rid of that pesky fraction by multiplying both sides of the equation by (x - 2):
2(x - 2) = x + 17
Expanding:
2x - 4 = x + 17
Moving things around a bit:
2x - x = 17 + 4
x = 21
So the solution to the equation is x = 21. Ta-da!
To solve the equation log2(x + 17) - log2(x - 2) = 1, we can use properties of logarithms.
The first property states that loga(b) - loga(c) = loga(b/c).
Applying this property to the given equation, we have:
log2(x + 17) - log2(x - 2) = log2[(x + 17)/(x - 2)] = 1
Now, we can rewrite the equation in exponential form:
2^1 = (x + 17)/(x - 2)
Simplifying, we have:
2 = (x + 17)/(x - 2)
Next, we can cross-multiply and solve for x:
2(x - 2) = x + 17
2x - 4 = x + 17
2x - x = 17 + 4
x = 21
Therefore, the solution to the equation log2(x + 17) - log2(x - 2) = 1 is x = 21.
To solve the logarithmic equation, we will use the properties of logarithms.
Step 1: Apply the quotient rule of logarithms. According to the quotient rule, log(base b)(a) - log(base b)(c) = log(base b)(a/c).
Applying the quotient rule to the given equation, we can rewrite it as:
log2((x + 17)/(x - 2)) = 1
Step 2: Convert the logarithmic equation to exponential form. In exponential form, the equation would be:
2^1 = (x + 17)/(x - 2)
Simplifying the equation:
2 = (x + 17)/(x - 2)
Step 3: Solve the equation by cross-multiplication.
2(x - 2) = x + 17
2x - 4 = x + 17
Step 4: Solve for x.
2x - x = 17 + 4
x = 21
Therefore, the solution to the logarithmic equation log2(x + 17) - log2(x - 2) = 1 is x = 21.