Solve for x:
log base 25 (log base 7 X)= -1/2
log25(log7(x)) = -1/2
since 25^1/2 = 5,
25^-1/2 = 1/5
log7(x) = 1/5
x = 7^1/5
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To solve for x in the equation:
log base 25 (log base 7 x) = -1/2
We'll start by applying the logarithmic properties and rules to simplify the equation.
1. Exponentiate both sides with base 25:
25^(log base 25 (log base 7 x)) = 25^(-1/2)
On the left side of the equation, the logarithm and exponent cancel out, leaving us with:
log base 7 x = 1/√25
2. Simplify the right side:
1/√25 can be simplified as 1/5 since the square root of 25 is 5:
log base 7 x = 1/5
3. Convert the logarithmic equation into an exponential equation:
Rewriting the equation in exponential form, we have:
7^(1/5) = x
4. Evaluate the right side:
Using a calculator, we determine that 7^(1/5) is approximately equal to 1.5157.
Therefore, x ≈ 1.5157.
So, the solution for x in the equation log base 25 (log base 7 x) = -1/2 is approximately x = 1.5157.