1/log_2(X) + 1/log_3(x) + 1/log_4(x) +1/log_5(x) =log_5(625)
To solve the equation 1/log_2(x) + 1/log_3(x) + 1/log_4(x) + 1/log_5(x) = log_5(625), we need to simplify the equation and find the value of x.
Step 1: Observe the equation and try to simplify the terms.
The equation has four terms, each involving different logarithms of x. To simplify each term, we can apply the logarithmic property that states log_b(x) = log_k(x) / log_k(b) for any base k.
Using this property, we can rewrite the equation as:
1/(log_2(x) / log_2(10)) + 1/(log_3(x) / log_3(10)) + 1/(log_4(x) / log_4(10)) + 1/(log_5(x) / log_5(10)) = log_5(625)
Rearranging the equation:
log_2(10) / log_2(x) + log_3(10) / log_3(x) + log_4(10) / log_4(x) + log_5(10) / log_5(x) = log_5(625)
Step 2: Simplify each term.
The logarithms with the same bases have a consistent denominator, so we can combine the terms:
(log_2(10) + log_3(10) + log_4(10) + log_5(10)) / log_2(x) = log_5(625)
Now, we can evaluate the numerator:
log_2(10) = log_2(2 * 5) = log_2(2) + log_2(5) = 1 + log_2(5)
log_3(10) = log_3(2 * 5) = log_3(2) + log_3(5)
log_4(10) = log_4(2 * 5) = log_4(2) + log_4(5)
log_5(10) = log_5(2 * 5) = log_5(2) + log_5(5) = 1 + log_5(5)
Our equation becomes:
(1 + log_2(5) + log_3(2) + log_3(5) + log_4(2) + log_4(5) + 1 + log_5(5)) / log_2(x) = log_5(625)
Simplifying inside the numerator:
(log_2(5) + log_3(2) + log_3(5) + log_4(2) + log_4(5) + log_5(5) + 1 + 1) / log_2(x) = log_5(625)
(log_2(5) + log_3(2) + log_3(5) + log_4(2) + log_4(5) + log_5(5) + 2) / log_2(x) = log_5(625)
Step 3: Simplify the right side of the equation.
Since log_5(625) = 4 (because 5^4 = 625), we can replace log_5(625) with 4:
(log_2(5) + log_3(2) + log_3(5) + log_4(2) + log_4(5) + log_5(5) + 2) / log_2(x) = 4
Step 4: Solve for x.
To isolate log_2(x), we can multiply both sides of the equation by log_2(x):
log_2(x) * [(log_2(5) + log_3(2) + log_3(5) + log_4(2) + log_4(5) + log_5(5) + 2) / log_2(x)] = 4 * log_2(x)
The log_2(x) cancels out on the left side:
log_2(5) + log_3(2) + log_3(5) + log_4(2) + log_4(5) + log_5(5) + 2 = 4 * log_2(x)
Now, we can simplify the equation further. However, it is important to note that there is no straightforward way to solve this equation algebraically. We need to make use of numeric methods such as approximation or iteration to find the value of x.