find the exact solution algebraically and check it by substituting into the orginal equation.
1. 36(1/3)^x/5=4
2. 32(1/4)^x/3=2
the way you typed it .....
36(1/3)^x = 20
(1/3)^x = .5555555..
x ln(1/3) = .55555..
x = .535 appr.
what you probably meant:
36(1/3)^(x/5) = 4
(1/3)^(x/5) = .111111..
x/5 = ln .111111 / ln(1/3)
x = 10
in the second, you probably have the same typo, and I will assume:
32(1/4)^(x/3) = 2
here is another way to do this:
notice that all factors are powers of 2, so ...
(2^5)(2^-2)^(x/3) = 2^1
2^(-2x/3) = 2^-4
-2x/3 = -4
x = 6
we could have done the first the same way, after dividing both sides by 32 to get
(1/3)^(x/5) = 1/9 = (1/3)^2
so x/5 = 2
x = 10
To find the exact solution algebraically and check it by substitution, we will solve the equations step by step.
1. Equation: 36(1/3)^(x/5) = 4
Step 1: Divide both sides by 36 to isolate the exponent term:
(1/3)^(x/5) = 4/36
Simplifying further:
(1/3)^(x/5) = 1/9
Step 2: Take the logarithm (base 1/3) of both sides to eliminate the exponent:
logbase(1/3)[(1/3)^(x/5)] = logbase(1/3)[1/9]
By the properties of logarithms, the left side simplifies to:
(x/5) = logbase(1/3)[1/9]
Step 3: Solve for x by multiplying both sides by 5:
x = 5 * logbase(1/3)[1/9]
Now, we have the exact solution for x in terms of logarithms.
To check the solution, substitute the value of x back into the original equation:
36(1/3)^(x/5) = 4
Substituting x = 5 * logbase(1/3)[1/9]:
36(1/3)^(5 * logbase(1/3)[1/9]/5) = 4
Simplify further if possible.
2. Equation: 32(1/4)^(x/3) = 2
Step 1: Divide both sides by 32 to isolate the exponent term:
(1/4)^(x/3) = 2/32
Simplifying further:
(1/4)^(x/3) = 1/16
Step 2: Take the logarithm (base 1/4) of both sides to eliminate the exponent:
logbase(1/4)[(1/4)^(x/3)] = logbase(1/4)[1/16]
By the properties of logarithms, the left side simplifies to:
(x/3) = logbase(1/4)[1/16]
Step 3: Solve for x by multiplying both sides by 3:
x = 3 * logbase(1/4)[1/16]
Now, we have the exact solution for x in terms of logarithms.
To check the solution, substitute the value of x back into the original equation:
32(1/4)^(x/3) = 2
Substituting x = 3 * logbase(1/4)[1/16]:
32(1/4)^(3 * logbase(1/4)[1/16]/3) = 2
Simplify further if possible.