solve log(5)(5x-4) = log(√5)(2-x)
log(5)(5x-4) = log(√5)(2-x) = log(5)[(2-x)^2]
5x-4=x^2-4x+4
why log(√5)(2-x)= log(5)[(2-x)^2]??
To solve the equation log(5)(5x-4) = log(√5)(2-x), we can use the property of logarithms that states: log(a)(m) = log(b)(n) if and only if m = n.
In this case, we have log(5)(5x-4) = log(√5)(2-x).
Since the bases of the logarithms are different (5 and √5), we need to transform them into the same base to equate the expressions inside the logarithms.
To do this, we can use the change of base formula, which states: log(a)(m) = log(b)(m) / log(b)(a).
Using this formula, we can rewrite the equation with a common base as:
log(5)(5x-4) = log(2-x) / log(√5)
Next, simplify the equation:
log(5)(5x-4) = log(2-x) / log(5)^(1/2)
To eliminate the logarithm on the right side, we can use the property log(a)(m^n) = n * log(a)(m):
log(5)(5x-4) = [log(2-x) / log(5)] * log(5)^(1/2)
Now, simplify further:
log(5)(5x-4) = [log(2-x) / log(5)] * log(5)^(1/2)
We can simplify log(5)^(1/2) by expressing it as a power of 5:
log(5)^(1/2) = log(5)^(1/2 * log(5)) = log(5)^(1/2 * 1) = log(5)^(1/2) = 1/2
Substituting this back into the equation, we get:
log(5)(5x-4) = [log(2-x) / log(5)] * 1/2
Multiplying both sides of the equation by log(5) to eliminate the logarithm on the left side, we have:
log(5)(5x-4) * log(5) = [log(2-x) / log(5)] * 1/2 * log(5)
Simplifying further:
(5x-4) * log(5) = [log(2-x) / log(5)] * 1/2 * log(5)
Now, cancel out the common factors of log(5) on both sides of the equation:
5x - 4 = [log(2-x) / log(5)] * 1/2
To simplify further, multiply both sides of the equation by 2:
2(5x - 4) = log(2 - x) / log(5)
Distribute the 2 on the left side:
10x - 8 = log(2 - x) / log(5)
To eliminate the fraction, multiply both sides of the equation by log(5):
(10x - 8) * log(5) = log(2 - x)
Now, we have:
10xlog(5) - 8log(5) = log(2 - x)
Finally, solve for x by isolating it on one side:
10xlog(5) = log(2 - x) + 8log(5)
Divide both sides of the equation by 10log(5):
x = (log(2 - x) + 8log(5)) / (10log(5))
If you have specific values for log(5) and log(2 - x), you can substitute them into the equation to find the value of x.