Solve
log10(8 x) − log10(1 + √x) = 2
for x.
log( 8x/(1+√x) = 2
8x/(1+√x) = 100
8x = 100 + 100√x
8x - 100 = 100√x
square both sides
64x^2 - 1600x + 10000 = 10000x
64x^2 - 11600x + 10 000 = 0
x = (11600 ± √132000000)/128
= 180.384 or .8662
since we squared we must check the answer
1. x = 180.384
LS = 2
RS = 2
2. x = .8662
LS = .555 ≠ RS
so x = 180.384
the 10s were supposed to be subscripts so just ignore them
If the 10's are subscripts, they are meant to be the base of the log, and are important.
log10(8 x) − log10(1 + √x) = 2
log10(8x/(1+√x)) = 2
Take anti-log (or raise to power of 10)
8x/(1+√x) = 10²
8x=100(1+√x)
If you plot the graph 8x-100(1+√(x)), you will find a minimum of about -400 at x=40, and a zero at about x=180.4.
To solve the equation log10(8x) - log10(1 + √x) = 2 for x, we can use logarithmic properties to simplify the equation and then solve for x.
Step 1: Apply the quotient rule of logarithms.
log10(8x) - log10(1 + √x) = 2
Step 2: Rewrite the equation using the logarithmic property.
log10(8x / (1 + √x)) = 2
Step 3: Rewrite 2 as log10(100), since 10^2 = 100.
log10(8x / (1 + √x)) = log10(100)
Step 4: Set the logarithmic expressions equal to each other.
8x / (1 + √x) = 100
Step 5: Multiply both sides of the equation by (1 + √x) to eliminate the denominator.
8x = 100(1 + √x)
Step 6: Distribute 100 to both terms on the right side.
8x = 100 + 100√x
Step 7: Move all terms to one side of the equation to set it to zero.
8x - 100 - 100√x = 0
Step 8: Factor out common terms.
8x - 100 - 100√x = 0
Step 9: Rearrange the equation to isolate the radical expression.
-100 - 100√x = -8x
Step 10: Move all terms containing x to one side.
-8x - 100√x - 100 = 0
Now, we have a quadratic equation in terms of √x, which we can solve using various methods such as factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula.
Step 11: Identify the coefficients of the quadratic equation.
a = -8
b = -100
c = -100
Step 12: Apply the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
x = (-(-100) ± √((-100)^2 - 4(-8)(-100))) / (2(-8))
x = (100 ± √(10000 - 3200)) / (-16)
x = (100 ± √(6800)) / (-16)
Step 13: Simplify the square root expression.
x = (100 ± √(4 * 1700)) / (-16)
x = (100 ± 2√(1700)) / (-16)
Now, we have two possible values for x:
x1 = (100 + 2√(1700)) / (-16)
x2 = (100 - 2√(1700)) / (-16)
These are the solutions to the given equation.