Find the sum of all value(s) of b such that 11/[log_2 (x)] + 1/[2 • log_25 (x)] - 3/[log_8 (x)] = 1/[log_b (x)] for all x > 1.
log_b (x), which means "the logarithm of x, base b."
11/[log_2 (x)] + 1/[2 • log_25 (x)] - 3/[log_8 (x)] = 1/[log_b (x)]
11/log2x + 1/[2log25x - 3/log8x = 1/logbx
recall that logcd = log d/log e , base 10 for both
so our equation becomes
11log2/logx + (1/2)log25/logx - 3log8/logx = logb/logx
multiply every term by logx
11log2 + (1/2)log25 - 3log8 = logb
log(2^11) + log(25^(1/2)) - log8^3 = logb/logx
log(2^11 * 5 / 8^3) = logb
log 20 = log b
b = 20
check my arithmetic
good answer. It might have been shorter just to note that
logbx = logxb
Saves going through the change of base. But that's always a good step to know, anyway.
To find the sum of all values of b that satisfy the equation, let's break down the given equation step by step.
First, let's manipulate the equation to simplify it:
11/[log₂(x)] + 1/[2 • log₂₅(x)] - 3/[log₈(x)] = 1/[log_b(x)]
The base of each logarithm term is different, so let's express them all in terms of the same base.
We know that logₘ(n) = log_b(n) / log_b(m), where log_b(n) represents the logarithm of n with base b.
Using this property, we can rewrite the equation:
11/[log₂(x)] + 1/[2 • log₂(x) / log₂(25)] - 3/[log₂(x) / log₂(8)] = 1/[log_b(x)]
Simplify further:
11/[log₂(x)] + 1/[log₂(x) + log₂(25)] - 3/[log₂(x) + log₂(8)] = 1/[log_b(x)]
Now, to add fractions with different denominators, we need to find a common denominator.
The common denominator here is log₂(x) • (log₂(x) + log₂(25)) • (log₂(x) + log₂(8)).
Multiplying both sides of the equation by this common denominator, we get:
11 • log₂(x) • (log₂(x) + log₂(8)) + (log₂(x) + log₂(25)) + 3 • (log₂(x) + log₂(8)) = log₂(x) • (log₂(x) + log₂(25)) • (log₂(x) + log₂(8)) / [log_b(x)]
Now, let's simplify the equation:
11 • log₂(x)² + 11 • log₂(x) • log₂(8) + log₂(x) + log₂(25) + 3 • log₂(x) • log₂(8) + 3 • log₂(8) = log₂(x) • (log₂(x) + log₂(25)) • (log₂(x) + log₂(8)) / [log_b(x)]
Combine like terms:
11 • log₂(x)² + 14 • log₂(x) • log₂(8) + log₂(x) + log₂(25) + 3 • log₂(8) = log₂(x) • (log₂(x) + log₂(25)) • (log₂(x) + log₂(8)) / [log_b(x)]
Now, if this equation holds true for all x > 1, then the coefficients of the logarithmic terms, on both sides of the equation, must be equal.
Comparing the coefficients:
11 = log₂(x) • (log₂(x) + log₂(25)) • (log₂(x) + log₂(8)) / [log_b(x)]
Now, to find the sum of all values of b, we need to solve this equation for b.
Since b is the base of the logarithm, it must be a positive real number.
We can take the reciprocal of both sides of the equation:
[log_b(x)] / [log₂(x) • (log₂(x) + log₂(25)) • (log₂(x) + log₂(8))] = 1 / 11
Cross-multiplying, we get:
11 • [log_b(x)] = [log₂(x)] • (log₂(x) + log₂(25)) • (log₂(x) + log₂(8))
Now, we can rewrite this equation in terms of b:
11/[log_b(x)] = [log₂(x)] • (log₂(x) + log₂(25)) • (log₂(x) + log₂(8))
To find the sum of all values of b, we would need the values of x.
Without the specific value(s) of x, it is not possible to directly find the sum of all values of b that satisfy the equation.