Solve for x.
log3(x+7)=2-log3(x-1)
Write the exact answer using base-10 logarithms.
log3(x+7)=2-log3(x-1)
log3(x+7) + log3(x-1) = 2
log3 [(x+7)/(x-1)] = 2
(x+7)(x-1) = 3^2
x^2 + 6x - 7 = 9
x^2 + 6x - 16 = 0
(x+8)(x-2) = 0
x = -8 or x = 2
but x=-8 would make log3(x+7) undefined, so
x = 2
( why would it ask to use base 10 logs ????)
To solve for x in the equation log3(x+7) = 2 - log3(x-1), we can start by simplifying the equation and then using logarithmic properties.
First, let's simplify the equation using the properties of logarithms. One property states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Applying this property to the right-hand side of the equation, we have:
log3(x+7) = 2 - log3(x-1)
log3(x+7) + log3(x-1) = 2
Next, we can use another logarithmic property which states that the sum of the logarithms of two numbers is equal to the logarithm of their product. Applying this property to the left-hand side of the equation, we have:
log3((x+7)(x-1)) = 2
Now, we have a single logarithm on both sides of the equation. The logarithm is in base 3, so we can rewrite the equation using base-10 logarithms. We can apply the change of base formula, which states that loga(b) = logc(b)/logc(a), where a, b, and c are positive real numbers and a and c are not equal to 1.
Let's rewrite the equation in base-10 logarithms:
log3((x+7)(x-1)) = 2
log((x+7)(x-1)) / log(3) = 2
To get rid of the logarithm on the left-hand side, we can exponentiate both sides with base 10:
10^(log((x+7)(x-1)) / log(3)) = 10^2
(x+7)(x-1) = 100
Now we have a quadratic equation. Let's expand the equation and simplify:
x^2 + 6x - 7 = 100
x^2 + 6x - 107 = 0
To solve this quadratic equation, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 6, and c = -107. Plugging these values into the quadratic formula, we get:
x = (-6 ± √(6^2 - 4(1)(-107))) / (2(1))
x = (-6 ± √(36 + 428)) / 2
x = (-6 ± √464) / 2
x = (-6 ± √(16 * 29)) / 2
x = (-6 ± 4√29) / 2
Simplifying further, we have:
x = (-3 ± 2√29)
So, the exact solutions for x in the equation log3(x+7) = 2 - log3(x-1) using base-10 logarithms are:
x = -3 + 2√29
x = -3 - 2√29