Question

Solve log subscript 2 baseline left parenthesis log subscript 5 baseline x right parenthesis equals 3.
(1 point)
Responses

x = 1,953,125
x = 1,953,125

x = 32,768
x = 32,768

x = 390,625
x = 390,625

x = 59,049

Answer 1

To solve the equation \( \log_2(\log_5 x) = 3 \), we can follow these steps:

1. Convert the equation from logarithmic form to exponential form:
\[
\log_5 x = 2^3 = 8
\]

2. Now, we can rewrite \( \log_5 x = 8 \) in exponential form:
\[
x = 5^8
\]

3. Calculate \( 5^8 \):
\[
5^8 = (5^4)^2 = 625^2
\]
Where \( 5^4 = 625 \).

4. Now calculate \( 625^2 \):
\[
625 \times 625 = 390625
\]

Therefore, \( x = 390625 \).

The solution to the equation \( \log_2(\log_5 x) = 3 \) is:
\[
\boxed{390625}
\]