Solve for the value of x: log5(x-2)+log8(x-4)=Log6(x-1)..For your information that 5,8 and 6 are bases not just multipliers. Thank you.
log 5(x-2) + log 8(x-4) = log 6(x-1)
log [5(x-2)*8(x-4)] = log 6(x-1)
so, if the logs are =, so are the expressions, and we have
40(x-2)(x-4) = 6(x-1)
20(x-2)(x-4) - 3(x-1) = 0
20x^2 - 123x + 163 = 0
now just use the quadratic formula to see the solutions. Check to be sure the values are defined in the original expressions.
6log(x^2+1)-x=0
To solve the equation log5(x-2) + log8(x-4) = log6(x-1) for the value of x, we can use the properties of logarithms to simplify the equation.
First, we can use the logarithmic identity log(a) + log(b) = log(a * b) to combine the two logarithms on the left side of the equation:
log5(x-2) + log8(x-4) = log40(x-2)(x-4)
Next, we can rewrite the equation using a common base. Since all three bases (5, 8, and 6) are different, we need to convert them to a common base. One way to do this is to express the bases in terms of a base that they can all be converted to. In this case, we can convert them to base 10:
log40(x-2)(x-4) = log10(x-2)(x-4) / log10(40)
Now we can simplify the equation further by using the logarithmic identity log(a) - log(b) = log(a / b) to rewrite the right side of the equation:
log40(x-2)(x-4) = log10(x-2)(x-4) - log10(40)
The equation now becomes:
log40(x-2)(x-4) = log10[(x-2)(x-4) / 40]
Next, we can use the logarithmic identity log(a^n) = n*log(a) to simplify the equation:
log40(x-2) + log40(x-4) = log10[(x-2)(x-4) / 40]
Now, we can apply the logarithmic property log(a) - log(b) = log(a / b) to combine the two logarithms on the left side of the equation:
log40[(x-2)(x-4)] = log10[(x-2)(x-4) / 40]
Finally, we can equate the two logarithmic expressions:
[(x-2)(x-4)] = [(x-2)(x-4) / 40]
Now we can solve for x. To do so, we can simplify the equation further by multiplying both sides by 40:
40[(x-2)(x-4)] = (x-2)(x-4)
Expanding both sides of the equation gives:
40x^2 - 320x + 480 = x^2 - 6x + 8
Now we can move all terms to one side of the equation to obtain a quadratic equation:
39x^2 - 314x + 472 = 0
This is a quadratic equation that can be solved using factoring, the quadratic formula, or completing the square, depending on the level of complexity desired.
Once we have solved the quadratic equation, we will obtain two possible values for x.