Math Question on Exponents!?

2^x = .25

Log(base 10) of 2^x = Log(base 10) of .25

short hand Log(base 10) of something to just Log(something)

Log(2^x) = Log(.25)

x * Log(2)= Log(.25)

this step comes from a property of Logarithms

x = Log(.25)/Log(2)

x = -2

Log(.4 ^x) = Log(125/8)

x * Log(.4) = Log(125/8)

x = Log(125/8)/Log(.4)

x = -3

0.25 is the same as 1/4 or 1/(2^2)

so 2^(-2)=0.25

0.4 = 2/5

0.4^(-1) = 5/2 and you would have to cube that to get 125/8

so, 0.4^(-3) = 125/8

2^x=0.25

Multiple both sides by natural log

ln(2^x)=ln(0.25)

This allows the x to be moved down in front.

xln(2)=ln(0.25)

Divide both sides by ln(2)

x=ln(0.25)/ln(2)

Plug it into a calculator for a decimal answer if you need.

The other one is solved the exact same way, just follow those steps.