What is the value of log7 log7 √(7√(7√7)) equal to?

Concept

A nested radical of the form √(a·√(a·√a)) can be rewritten as a single power of a by converting each radical into a fractional exponent and adding the exponents, since each further radical layer multiplies the running exponent by 1/2. Once the expression is a pure power of the base, the logarithm identities logb(bk) = k, logb(m/n) = logb m − logb n, and logb(nk) = k·logb n bring it to a closed form.

Application

1. Write every radical as a fractional power of 7: √7 = 71/2; the middle radical is √(7 · 71/2) = √(73/2) = 73/4.

2. The outer radical is then √(7 · 73/4) = √(77/4) = 77/8, so the whole nested radical √(7√(7√7)) equals 77/8.

3. Apply the inner logarithm: log7(77/8) = 7/8, using logb(bk) = k.

4. Apply the outer logarithm to this result: the expression becomes log7(7/8).

5. Split the quotient: log7(7/8) = log77 − log78 = 1 − log78.

6. Rewrite 8 as 23 and apply the power rule: log78 = log7(23) = 3 log72.

7. Substitute back: log7(7/8) = 1 − 3 log72.

Cross-check

Numerically, 77/8 ≈ 5.75, and log7(5.75) ≈ 0.875 = 7/8, confirming the inner step. Then log7(0.875) ≈ −0.0686, and 1 − 3 log72 ≈ 1 − 3(0.3562) ≈ 1 − 1.0686 ≈ −0.0686 as well — the two routes agree.

So the value of log7 log7 √(7√(7√7)) is 1 − 3 log72.