can you show me how to do these problems solve the equation by finding the exact solution
1/3x-2=2 b) 1/3log(x)-log(100)=2
c)ln(x)-ln(5)=7
Sure! Let's break down each problem step by step.
a) To solve the equation 1/3x - 2 = 2:
Step 1: Add 2 to both sides of the equation to isolate the variable on one side.
1/3x - 2 + 2 = 2 + 2
1/3x = 4
Step 2: In order to get rid of the fraction, we can multiply both sides of the equation by the reciprocal of 1/3, which is 3/1.
(1/3x) * (3/1) = 4 * (3/1)
x = 12
So the exact solution to the equation 1/3x - 2 = 2 is x = 12.
b) To solve the equation 1/3log(x) - log(100) = 2:
Step 1: Combine the logarithms using the properties of logarithms. The subtraction of logarithms is the same as taking the logarithm of the division of the arguments:
log(x^(1/3)) - log(100) = 2
Step 2: Apply the quotient rule of logarithms, which states that log(a/b) = log(a) - log(b):
log(x^(1/3) / 100) = 2
Step 3: Rewrite the equation using exponential form. The logarithm function is essentially the inverse of the exponential function:
x^(1/3) / 100 = 10^2
Step 4: Simplify the right side of the equation:
x^(1/3) / 100 = 100
Step 5: Multiply both sides of the equation by 100 to eliminate the fraction:
x^(1/3) = 100 * 100
x^(1/3) = 10000
Step 6: Raise both sides of the equation to the power of 3 to remove the exponent:
(x^(1/3))^3 = 10000^3
x = 100000000
So the exact solution to the equation 1/3log(x) - log(100) = 2 is x = 100,000,000.
c) To solve the equation ln(x) - ln(5) = 7:
Step 1: Combine the logarithms using the properties of logarithms. The subtraction of logarithms is the same as taking the logarithm of the division of the arguments:
ln(x / 5) = 7
Step 2: Rewrite the equation using exponential form. The natural logarithm function (ln) is essentially the inverse of the exponential function with base e:
e^(ln(x / 5)) = e^7
Step 3: Simplify the equation and eliminate the natural logarithm:
x / 5 = e^7
Step 4: Multiply both sides of the equation by 5 to isolate the variable on one side:
x = 5 * e^7
So the exact solution to the equation ln(x) - ln(5) = 7 is x = 5 * e^7, where e is the base of the natural logarithm and approximately equal to 2.71828.