Solve the simultaneous equation logx+logy=4
I would appreciate it if you help me
logx+logy=4
log(xy)=4
xy=10^4
Well, solving equations can be quite tricky, but I'll give it a shot!
To solve the simultaneous equation logx + logy = 4, let's use the properties of logarithms.
First, we can rewrite the equation using the logarithmic property that states log(a) + log(b) = log(ab).
So, log(x) + log(y) = 4 can be rewritten as log(xy) = 4.
Now, let's take both sides of the equation to the exponent to eliminate the logarithm:
e^(log(xy)) = e^4
Since raising e to the power of log base e cancels each other out, we're left with:
xy = e^4
Voila! We have solved the equation. But if you're looking for the specific values of x and y, I'm afraid we need some additional information or another equation to proceed.
To solve the simultaneous equation log(x) + log(y) = 4, we can use the properties of logarithms to simplify the equation and then solve for the variables.
Step 1: Combine the logarithms using the property log(a) + log(b) = log(ab):
log(xy) = 4
Step 2: Rewrite the equation using exponential form:
xy = 10^4
Step 3: Simplify the right side:
xy = 10,000
Now, the equation is xy = 10,000.
To find the solutions for x and y, you can either use trial and error or numerical methods. Some possible solutions are:
x = 1, y = 10,000
x = 2, y = 5,000
x = 4, y = 2,500
x = 5,000, y = 2
x = 10,000, y = 1
These are just a few examples, and there may be other possible solutions as well.
To solve the simultaneous equation logx + logy = 4, we can use the properties of logarithms. The first step is to combine the logs using the logarithmic identity:
logx + logy = log(xy)
So, the equation becomes:
log(xy) = 4
To solve for x and y, we need to convert the logarithmic equation into exponential form. In exponential form, the equation becomes:
xy = 10^4
Simplifying, we have:
xy = 10000
Now we have an equation with two variables. To solve for one variable, we need to express it in terms of the other variable. Let's solve for y:
y = 10000/x
Now, we can substitute this expression into one of the original equations to solve for x. Substituting into the first equation:
logx + log(10000/x) = 4
Using the logarithmic identity again:
log(x * 10000/x) = 4
Simplifying further:
log(10000) = 4
Since log(10000) = 4, the equation simplifies to:
4 = 4
This equation is true for any value of x. Therefore, x can take any positive real number.
To find y, substitute the value of x back into the equation y = 10000/x. For any positive value of x, y will also be a positive real number.
Therefore, the solution to the simultaneous equation logx + logy = 4 is x can be any positive real number, and y = 10000/x.