find exact solution algebraically and check it by substituting into the original equation.
2(10^-x/3)=20
logtiny4(x-5)=-1
you probably meant:
2(10^-(x/3) )=20
10^(-x/3) = 10 = 10^1 , now you have the same bases, so
-x/3 = 1
x = -3
log4 (x-5) = -1
by definition:
x-5 = 4^-1
x-5 = 1/4 = .25
x = 5.25
x = 1029
To find the exact solution algebraically, we will solve each equation step-by-step:
1. Equation: 2(10^(-x/3)) = 20
First, divide both sides of the equation by 2 to isolate the exponent term:
10^(-x/3) = 10
Next, apply base 10 logarithm (log10) to both sides of the equation, which cancels out the exponent:
log10[10^(-x/3)] = log10(10)
Using the property logb(b^x) = x, we get:
(-x/3) = 1
Now, multiply both sides by -3 to solve for x:
-3 * (-x/3) = -3
x = -3
To check the solution, substitute x = -3 back into the original equation:
2(10^(-(-3)/3)) = 20
2(10^(1)) = 20
2(10) = 20
20 = 20
Since both sides are equal, the solution x = -3 is correct.
2. Equation: logtiny4(x-5) = -1
To remove the logarithm, rewrite the equation in exponential form:
4^(-1) = x - 5
Simplify the exponent:
1/4 = x - 5
Now, add 5 to both sides to solve for x:
1/4 + 5 = x
5 1/4 = x
To check the solution, substitute x = 5 1/4 back into the original equation:
logtiny4[(5 1/4) - 5] = -1
logtiny4[(21/4) - 20/4] = -1
logtiny4(1/4) = -1
Using the property of logarithms, logb(b^x) = x, we know that logtiny4(4^(-1)) = -1:
-1 = -1
Since both sides are equal, the solution x = 5 1/4 is correct.