Find the value(s) of x for which the equation is true.
log(x + 20) = log(x) + log(20)
Solve the logarithmic equation for x. (Enter your answers as a comma-separated list. Round your answer to four decimal places.)
ln(x − 2) + ln(x + 3) = 1
x + 20 = 20 x
(x - 2)(x + 3) = e
To find the value(s) of x for which the equation log(x + 20) = log(x) + log(20) is true, we can simplify the equation and solve for x.
Using the logarithmic rule log(a) + log(b) = log(a * b), we can rewrite the equation as:
log(x + 20) = log(20x)
Since the logarithmic functions are equal, their inputs must also be equal. Therefore, we have:
x + 20 = 20x
Now, let's solve for x:
20 = 20x - x
20 = 19x
x = 20/19
So, the value of x for which the equation is true is x = 20/19.
Now, let's solve the logarithmic equation ln(x − 2) + ln(x + 3) = 1.
Using the logarithmic property ln(a) + ln(b) = ln(a * b), we can simplify the equation to:
ln((x - 2)(x + 3)) = 1
To remove the natural logarithm, we can rewrite the equation in exponential form:
e^1 = (x - 2)(x + 3)
Simplifying further, we have:
e = x^2 + x - 6
Rearranging the equation, we get:
x^2 + x - 6 - e = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
After solving the quadratic equation, we get two possible values of x. Let's assume they are x1 and x2.
The solution to the logarithmic equation ln(x − 2) + ln(x + 3) = 1 is given by the values of x1 and x2.
Rounded to four decimal places, the solution would be x1, x2 (if applicable).