Solve for x 8(4^(6-2x))+13=41

Problem

8*(4^(6−2*x))+13=41

Solution

1. Isolate the exponential term by subtracting 13 from both sides of the equation.

8*(4^(6−2*x))=28

2. Divide both sides by 8 to further isolate the base with the exponent.

4^(6−2*x)=28/8

3. Simplify the fraction on the right side by dividing the numerator and denominator by 4.

4^(6−2*x)=7/2

4. Apply the natural logarithm to both sides to move the variable out of the exponent.

ln(4^(6−2*x))=ln(7/2)

5. Use the power rule of logarithms to bring the exponent down as a multiplier.

(6−2*x)*ln(4)=ln(7/2)

6. Divide both sides by ln(4) to isolate the linear expression containing x

6−2*x=ln(7/2)/ln(4)

7. Subtract 6 from both sides of the equation.

−2*x=ln(7/2)/ln(4)−6

8. Divide by -2 to solve for x

x=(ln(7/2)/ln(4)−6)/(−2)

9. Simplify the expression by distributing the negative sign and the division.

x=3−ln(7/2)/(2*ln(4))

10. Rewrite the denominator using the property 2*ln(4)=ln(4^2)=ln(16)

x=3−ln(3.5)/ln(16)

Final Answer

x=3−ln(3.5)/ln(16)

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