SOLUTION: Solve. Round to four decimal places a) e^3x >_ 21 b) (1/5)^t-2 = 125

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Solve. Round to four decimal places a) e^3x >_ 21 b) (1/5)^t-2 = 125      Log On


   

Question 354347: Solve. Round to four decimal places
a) e^3x >_ 21
b) (1/5)^t-2 = 125
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
a) e%5E%283x%29+%3E=+21
Find the natural log (because of the "e" and because we want a decimal):
ln%28e%5E%283x%29%29+%3E=+21
Use a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent out in front:
3x%2Aln%28e%29+%3E=+21
By definition, ln(e) = 1 so this simplifies to:
3x+%3E=+21
Divide both sides by 3:
x+%3E=+21

b) %281%2F5%29%5E%28t-2%29+=+125
The quick way to solve this comes from recognizing that both sides are powers of 5: 1%2F5+=+5%5E%28-1%29 and 125+=+5%5E3. Subsituting these in we get:
%285%5E%28-1%29%29%5E%28t-2%29%29+=+5%5E3
On the left side the rule is to multiply exponents:
5%5E%28-t%2B2%29+=+5%5E3
Now the equation explicitly says each side is a power of 5. And if these two powers of 5 are equal, then the exponents must be equal!
-t+%2B+2+=+3
Solving this:
-t+=+1
t+=+-1