solve the equation

Question

(25/9)^x+1 = (3/5)^{x-1         /=devided by}

Answer 1

Hi Melinda,

 

This is an exponential equation because the variable is in the exponent. One way to solve this type of equation is to take the logarithm of both sides and use properties of logarithms to isolate the variable. It is possible in some cases to use the fact that if powers are equal and have the same base, then the exponents must be equal. Can we transform these equal powers to have the same base?

 

First of all notice that 25/9 can be written as (5/3)² and if we use the reciprocal of 5/3 we get (3/5)^-2, using the definition of negative exponent. 

 

So now we have:

 

[(3/5)^-2]^x+1 = (3/5)^x-1

 

(3/5)^-2x-2 = (3/5)^x-1  (using the power of a power law of exponents--multiply -2(x+1))

 

Finally the bases of these powers are equal, so the exponents must be equal.

 

-2x-2 = x-1

-1 = 3x

x=-1/3

 

You can check that this if fact is the correct answer by substituting -1/3 for x in the original equation.

See https://www.desmos.com/calculator/eqzkj10vdz for an indication.