Frustrating exponential problem

Question

I have tried for hours and cannot figure out how to solve the following problem:

(5^t) times (4)^2t = 10.     Solve for "t"

Can someone please help me?

Answer 1

I would use logarithms.  Log both sides of the equation.

 

Log[(5^t)(4)^2t] = Log10

 

 

Then using the properties of log,

 

Log(5^t) + Log(4)^2t = Log10

 

Log(5^t) + Log(4^t) + Log(4^t )= Log10

 

Log(5^t) + 2Log(4^t)= Log10

 

tLog(5) + 2tLog(4) = 1

 

 

Factor out the t on the left side of equation.

 

t[Log(5) + 2Log(4)] = 1

 

 

t = 1 / [Log(5) + 2Log(4)]