Solve for x (1/4)^x=75

Problem

(1/4)x=75

Solution

1. Apply the logarithm to both sides of the equation to isolate the exponent. We will use the natural logarithm ln()

ln((1/4)x)=ln(75)

2. Use the power rule for logarithms, which states ln(ab)=b*ln(a) to move the variable x in front of the logarithm.

x*ln(1/4)=ln(75)

3. Isolate the variable x by dividing both sides of the equation by ln(1/4)

x=ln(75)/ln(1/4)

4. Simplify the expression using the property ln(1/a)=−ln(a) to rewrite the denominator.

x=ln(75)/(−ln(4))

5. Rewrite the result to provide the exact value of x

x=−ln(75)/ln(4)

Final Answer

x=−ln(75)/ln(4)

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