Solve for x 6(2^(3x-1))-7=9

Problem

6*(2(3*x−1))−7=9

Solution

1. Isolate the term containing the exponent by adding 7 to both sides of the equation.

6*(2(3*x−1))=16

2. Divide both sides by 6 to further isolate the exponential expression.

2(3*x−1)=16/6

3. Simplify the fraction on the right side by dividing the numerator and denominator by 2

2(3*x−1)=8/3

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

ln(2(3*x−1))=ln(8/3)

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

(3*x−1)*ln(2)=ln(8/3)

6. Divide both sides by ln(2) to isolate the linear expression in x

3*x−1=ln(8/3)/ln(2)

7. Add 1 to both sides of the equation.

3*x=ln(8/3)/ln(2)+1

8. Divide the entire expression by 3 to solve for x

x=(ln(8/3)/ln(2)+1)/3

9. Simplify the expression using logarithm properties, noting that ln(8/3)/ln(2)=(log_2)(8/3)

x=((log_2)(8/3)+1)/3

10. Expand the logarithm using the quotient rule (log_2)(8/3)=(log_2)(8)−(log_2)(3)

x=(3−(log_2)(3)+1)/3

11. Combine the constants in the numerator.

x=(4−(log_2)(3))/3

Final Answer

x=(4−(log_2)(3))/3

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