Solve for x 2^(x-1)=20

Problem

2(x−1)=20

Solution

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

ln(2(x−1))=ln(20)

2. Use the power rule of logarithms, which states ln(ab)=b*ln(a) to bring the exponent down.

(x−1)*ln(2)=ln(20)

3. Isolate the term containing x by dividing both sides by ln(2)

x−1=ln(20)/ln(2)

4. Solve for x by adding 1 to both sides of the equation.

x=ln(20)/ln(2)+1

5. Simplify the expression using the change of base formula ln(20)/ln(2)=(log_2)(20)

x=(log_2)(20)+1

6. Alternatively simplify by rewriting ln(20) as ln(2⋅5)=2*ln(2)+ln(5)

x=(2*ln(2)+ln(5))/ln(2)+1

x=2+ln(5)/ln(2)+1

x=3+ln(5)/ln(2)

Final Answer

x=ln(20)/ln(2)+1

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