1. Solve for x:log27 5x-log27(x+2)=2/3
log27 5x-log27(x+2)=2/3
assuming you have base 27 in your log
then:
log27 (5x/(x+2)) = 2/3 , where x > 0 by the definition for logs
5x/(x+2) = 27^(2/3)
5x/(x+2) = 9
9x + 18 = 5x
4x = -18
x = -9/2 , but x > 0
so there is no solution.
To solve the equation log27 5x - log27 (x+2) = 2/3 for x, we can use the properties of logarithms. Here's the step-by-step process:
Step 1: Use the logarithmic identity logb(a) - logb(b) = logb(a/b):
log27 5x/(x+2) = 2/3
Step 2: Convert the equation into exponential form:
27^(2/3) = 5x/(x+2)
Step 3: Simplify the left side of the equation:
(27^(1/3))^2 = 5x/(x+2)
27^(1/3) = 5x/(x+2)
Step 4: Cube both sides to eliminate the fractional exponent:
(27^(1/3))^3 = (5x/(x+2))^3
27 = (5x)^3 / (x+2)^3
27 = 125x^3 / (x+2)^3
Step 5: Cross-multiply:
27(x+2)^3 = 125x^3
Step 6: Expand the left side of the equation:
27(x+2)(x+2)(x+2) = 125x^3
Step 7: Simplify and multiply:
27(x+2)^3 = 125x^3
Step 8: Expand the left side of the equation:
27(x^3 + 6x^2 + 12x + 8) = 125x^3
Step 9: Distribute and simplify:
27x^3 + 162x^2 + 324x + 216 = 125x^3
Step 10: Rearrange the equation:
27x^3 - 125x^3 + 162x^2 + 324x - 216 = 0
Step 11: Combine like terms:
-98x^3 + 162x^2 + 324x - 216 = 0
Now, this is a cubic equation, and solving it can be quite complex and involve numerical methods or approximations. Therefore, it is challenging to provide an exact solution for x without using advanced mathematical techniques or software.
To solve the equation log27 5x - log27 (x+2) = 2/3, we can use the properties of logarithms to simplify the expression.
Step 1: Applying the quotient rule of logarithms: loga (b/c) = loga b - loga c, we rewrite the equation as:
log27 (5x/(x+2)) = 2/3
Step 2: Rewrite the equation using exponentiation form:
27^(2/3) = 5x/(x+2)
Step 3: Simplify the left side of the equation:
(3√27)^2 = 5x/(x+2)
Step 4: Simplify 27 to its prime factorization:
(3^3)^2 = 5x/(x+2)
Step 5: Simplify the exponents:
3^2 = 5x/(x+2)
Step 6: Simplify the left side of the equation:
9 = 5x/(x+2)
Step 7: Multiply both sides of the equation by (x+2) to eliminate the fraction:
9(x+2) = 5x
Step 8: Distribute the 9 on the left side:
9x + 18 = 5x
Step 9: Subtract 5x from both sides to isolate the variable term:
9x - 5x + 18 = 0
4x + 18 = 0
Step 10: Subtract 18 from both sides:
4x = -18
Step 11: Divide both sides by 4 to solve for x:
x = -18/4
Step 12: Simplify the fraction:
x = -9/2
Therefore, the solution to the equation log27 5x - log27 (x+2) = 2/3 is x = -9/2.