solve the exponential equation

Question

(1/4)^-2x=64

Answer 1

First simplify the left side:

 

(1/4)^-2x = [(1/4)^-2 ]^x 

 

Look at the term inside the brackets:

 

(1/4)^-2 = 1/ (1/4)² = 1 / (1/16) = 16

 

So the left side is 16^x.  But 16 is the same as 2⁴ so the left side can be written as 2^4x

 

The right side is 64, which is 2⁶

 

So the equation can be written as 2^4x = 2⁶

 

From this you can see that 4x = 6, so x = 6/4 = 3/2.

 

 

Answer 2

log_1/4[(1/4)^-2x] = log_1/4(64) 

-2x = log_1/4 64

-2x = log₁₀(64)/log₁₀(1/4) = log_e(64)/log_e(1/4) = -3

Thus x = 3/2 = 1.5

 

In fact we can check that (1/4)^-2(3/2) = (1/4)^-3 = 4³ = 64

 

Remember that to find logaB for any base a, we can always use the logs in base 10 or e as follows

log_aB = log₁₀B/log₁₀a

 

In this case it would have been even better to use logs in base 2. In fact, note that

 

log_1/4(64) = log₂(64)/log₂(1/4) = log₂(2⁶)/log₂(2^-2) = 6/-2 = -3  without even having to use the calculator!