solv log exponent x and base 2 + log exponent 2 base x
To solve the expression log₂(x) + logₓ(2), we will use the properties of logarithms.
First, let's rewrite the expression using the logarithm property logₐ(b) = 1 / log_b(a):
log₂(x) + logₓ(2) = 1 / log_x(2) + 1 / log_2(x)
Now, let's combine the two fractions with a common denominator:
1 / log_x(2) + 1 / log_2(x) = (log_2(x) + log_x(2)) / (log_x(2) * log_2(x))
Next, we can use the logarithm property log_b(a) + log_b(c) = log_b(a * c) to simplify the numerator:
log_2(x) + log_x(2) = log_2(x * 2) = log_2(2x)
Substituting this result back into the expression:
(log_2(x) + log_x(2)) / (log_x(2) * log_2(x)) = log_2(2x) / (log_x(2) * log_2(x))
Now, we can simplify further by noticing that log_b(b) = 1:
log_2(2x) / (log_x(2) * log_2(x)) = 1 / (log_x(2) * log_2(x))
So, the final expression is:
1 / (log_x(2) * log_2(x))
This is the simplified form of the given expression.