Log3^x+logx^3=2.5
Or, did you mean
log3x+logx3 = 2.5
?
In that case, recall that one log is the reciprocal of the other. So, If you let
u = log3x, then you have
u + 1/u = 5/2
2u^2 - 5u + 2 = 0
(2u-1)(u-2) = 0
u = 1/2 or u=2
Thus, u = √3 or 9
Log3^x+logx^3=2.5
xlog3 + 3logx = 2.5
nasty equation to solve
let's try this very primitive method:
let x = 1.5 , LS = 1.5log3 + 3log1.5 = 1.243, too small
let x = 1.8, LS = 1.62 , still too small
let x = 2.5, LS = 2.3866 , getting there
let x = 3, LS = 2.86 , too high
let x = 2.75 , LS = 2.63 , still too high
let x = 2.6, LS = 2.48 , getting closer
let x = 2.625, LS = 2.509 , just a bit too high
let x = 2.62, LS = 2.5049
let x = 2.61, LS = 2.495
let x = 2.615, LS = 2.500087
close enough
x = appr 2.615
there are of course more sophisticated methods, such as Newton's Method, but what I did is very easy to understand.
you mean x=sqrt3 or 9, not u
Log3x+logx3=2.5
To solve the equation log3^x + logx^3 = 2.5, we can use logarithmic properties to simplify and calculate x.
First, let's apply the product rule of logarithms, which states that the sum of two logarithms of the same base is equivalent to the logarithm of their product:
log3^x + logx^3 = log(3^x * x^3)
Next, we can simplify further using the power rule of logarithms:
= log(3^x * x^3)
= log(3^x * (x^2)^1 * x^1)
= log(3^x * x^(2+1) * x^1)
= log(3^x * x^3)
Now, we can express the equation in exponential form by taking the base 10 logarithm of both sides:
log(3^x * x^3) = 2.5
10^(log(3^x * x^3)) = 10^2.5
We know that 10^log(a) is equal to a, so the equation becomes:
3^x * x^3 = 100
At this point, we have a nonlinear equation that cannot be solved algebraically. However, we can use numerical methods or approximations to find an approximate solution.
One possible approach is to use trial and error or guess-and-check method. By trying different values of x, we can calculate the left-hand side of the equation until we find an approximate solution that satisfies the equation.
Another method is to use a graphing calculator or a computer program that can calculate the intersection point of the equations y = 3^x * x^3 and y = 100. This will give you the approximate value of x.
Remember, since this equation involves logarithms, it's always a good practice to check your solution(s) by plugging them back into the original equation to ensure they satisfy it.