Solve the following exponential equation. Give an exact answer in terms of the natural logarithm.

Question

Solve the following exponential equation. Give an exact answer in terms of the natural logarithm (other methods will not receive full credit) and a decimal rounded to 3 decimals. Show step by step work

3^1−7𝑥 = 7^𝑥

Answer 1

3^(1-7x)=7^x

ln3^(1-7x)=ln7^x

(1-7x)ln3=xln7

(1-7x)/x=ln7/ln3

(1-7x)/x=1.9459/1.0986

(1-7x)/x=1.77125

1-7x=1.77125x

1=7x+1.77125x

1=8.77125x

x=1/(8.77125)

x=0.114

check:

3^(1-.798)=7^(0.114)

3^(0.202)=7^(0.114)

1.24847≈1.24836