If Logx (1 / 8) = - 3 / 2, then x is equal to
log (1/8)/log x =-3/2 ;now log is base 10.
therefore,
log (1/8) = -0.9030.
then
log x = -0.6020
find the anti-log and you have
x = 0.25
logx (1/8) = -3/2
means:
x^(-3/2) = 1/8
x^(3/2) = 8
(√x)^3 = 8
√x = 2
x = 4
check:
log4 (1/8)
= log41 - log48
= 0 - 3/2 , (since 4^(3/2) = 8)
= -3/2
To find the value of x, we can rewrite the equation in exponential form. The logarithm equation:
Logₓ (1/8) = -3/2
can be rewritten as:
x^(-3/2) = 1/8
To solve for x, we will raise both sides of the equation to the reciprocal power of -3/2, which is -2/3:
(x^(-3/2))^(-2/3) = (1/8)^(-2/3)
Simplifying both sides:
x^(3/2 * -2/3) = (1/8)^(2/3)
x^(-1) = (1/8)^(2/3)
Now, we can simplify the right side of the equation:
(1/8)^(2/3) = (1^(2/3))/(8^(2/3))
(1/8)^(2/3) = 1/4
The equation now becomes:
x^(-1) = 1/4
To find x, we can take the reciprocal of both sides:
1/x = 1/4
Multiplying both sides by x:
1 = x/4
To isolate x, we multiply both sides by 4:
4 = x
Therefore, x is equal to 4.
To find the value of x, we need to use the properties of logarithms to manipulate the given equation. Since we have a logarithm with a base of x, we can rewrite the equation using the equivalent exponential form:
x^(-3/2) = 1/8
To simplify this equation, we can express 1/8 as a power of 2:
x^(-3/2) = 2^(-3)
Now, we can use the property of equality for exponents, which states that if two exponential expressions with the same base are equal, then their exponents must also be equal:
-3/2 = -3
We can multiply both sides of the equation by 2 to get rid of the fraction:
-3 * 2 = -3/2 * 2
-6 = -3
However, this equation is not true. So, there is no real value of x that satisfies the given equation Logx (1 / 8) = - 3 / 2.
Therefore, the equation has no solution for x.