Solve for x as an exact value:

Question

 

5^x=2^(x-4)?

Answer 1

5^x = 2^(x-4)

logarithm apply to each side

log 5^x = log 2^(x-4)

Formula : log Aⁿ = n log A

x log 5 = (x - 4) log 2

Simplify

x log 5 = x log 2 - 4 log 2

Subtract 'x log 2' from each side.

x log 5 - x log 2 = x log 2 - 4 log 2 + x log 2

x log 5 - x log 2 = - 4 log 2

Take out common factors.

x (log 5 - log 2) = - 4 log 2

Multiply each side by negative one.

x (log 2 - log 5) = 4 log 2

Note : log (A - B) = log (A/B) and nlog A = log Aⁿ

x log (2/5) = log 2⁴

x log (2/5) = log 16

Divide each side by 'log (2/5)'

[x log (2/5)] / [log (2/5)] = log 16 / log (2/5)

Simplify

x = log 16 / log (2/5)