5logxbase4+48log4basex=x/8
help me plz step
5logxbase4+48log4basex=x/8
--->
5 log 4 x + 48log x 4 = x/8
One of rules of logs:
log a b = 1/log b a
and furthermore:
log a b = log b/log a
so
5 log 4 x + 48log x 4 = x/8
I don't know of any nice method to solve this, I ran it through Wolfram and got x = 2130.13 , which satisfies the original equation
http://www.wolframalpha.com/input/?i=solve+5(log4)%2Flogx+%2B+48logx%2Flog4+%3D+x%2F8
My equation near the end should have included:
5 log 4 x + 48log x 4 = x/8
5 log4 x + 48/log4 x = x/2
or
5log 4/log x + 48log x/log 4 = x/2
btw, that is what I used in Wolfram , so my answer stands as is
If we let
u = log_4(x) then x = 4^u and we have
5u + 48/u = 4^u/8
the solution to this is u=4
so, x=4^u = 256
Steve how did you get u=4?
To solve the equation 5logₓ(base 4) + 48log₄(base x) = x/8, we can follow these steps:
Step 1: Simplify the equation.
Combine the logarithmic terms using logarithmic properties. Since both terms have a base of 4, we can rewrite the equation as:
5logₓ(4) + 48log₄(x) = x/8
Step 2: Change the base of the logarithms.
We can use the formula logₐ(b) = logₓ(b)/logₓ(a) to change the base of the logarithm.
Let's change the base of the first term, 5logₓ(4):
logₓ(4) = log₄(4)/log₄(x) = 1/log₄(x)
Now, the equation becomes:
5(1/log₄(x)) + 48log₄(x) = x/8
Step 3: Make a substitution.
Let's substitute a variable to simplify the equation further.
Let u = log₄(x)
Now, our equation is:
5(1/u) + 48u = x/8
Step 4: Solve the equation.
To solve the equation, we need to determine the value of u that satisfies it. Unfortunately, there is no straightforward algebraic method to find an exact solution. We will need to resort to numerical methods or the use of a graphing calculator to approximate the value.
One option is to use numerical methods like the bisection method or Newton's method to find the value of u that satisfies the equation. These methods involve making initial estimates and iteratively refining them until a solution is found. However, these methods can be complex to apply manually.
Another option is to graph the two sides of the equation separately and observe where they intersect. A graphing calculator or online graphing tool can help plot the two functions: y = 5(1/u) and y = x/8. The intersection point(s) will give an approximate solution.
Keep in mind that the equation may have multiple solutions, so it's important to carefully analyze the graph or use additional numerical methods to find them all.
I hope this explanation helps you in solving the equation!